physics 461 / quantum mechanics i
TRANSCRIPT
Physics 461 / Quantum Mechanics I
P.E. ParrisDepartment of Physics
University of Missouri-RollaRolla, Missouri 65409
January 10, 2005
CONTENTS
1 Introduction 71.1 What is Quantum Mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 What is Mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.2 Postulates of Classical Mechanics: . . . . . . . . . . . . . . . . . . 8
1.2 The Development of Wave Mechanics . . . . . . . . . . . . . . . . . . . . . 101.3 The Wave Mechanics of Schrödinger . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Postulates of Wave Mechanics for a Single Spinless Particle . . . . 131.3.2 Schrödinger’s Mechanics for Conservative Systems . . . . . . . . . 161.3.3 The Principle of Superposition and Spectral Decomposition . . . . 171.3.4 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.5 Superpositions of Plane Waves and the Fourier Transform . . . . . 23
1.4 Appendix: The Delta Function . . . . . . . . . . . . . . . . . . . . . . . . 26
2 The Formalism of Quantum Mechanics 312.1 Postulate I: Specification of the Dynamical State . . . . . . . . . . . . . . 31
2.1.1 Properties of Linear Vector Spaces . . . . . . . . . . . . . . . . . . 312.1.2 Additional Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 332.1.3 Continuous Bases and Continuous Sets . . . . . . . . . . . . . . . . 342.1.4 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.5 Expansion of a Vector on an Orthonormal Basis . . . . . . . . . . 382.1.6 Calculation of Inner Products Using an Orthonormal Basis . . . . 392.1.7 The Position Representation . . . . . . . . . . . . . . . . . . . . . 402.1.8 The Wavevector Representation . . . . . . . . . . . . . . . . . . . . 41
2.2 Postulate II: Observables of Quantum Mechanical Systems . . . . . . . . . 432.2.1 Operators and Their Properties . . . . . . . . . . . . . . . . . . . . 432.2.2 Multiplicative Operators . . . . . . . . . . . . . . . . . . . . . . . . 452.2.3 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.4 Ket-Bra Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.5 Projection Operators: The completeness relation . . . . . . . . . . 492.2.6 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2.7 Action of Operators on Bras of S∗ . . . . . . . . . . . . . . . . . . 522.2.8 Hermitian Conjugation . . . . . . . . . . . . . . . . . . . . . . . . 522.2.9 Hermitian, Anti-Hermitian, and Unitary Operators . . . . . . . . . 542.2.10 Matrix Representation of Operators . . . . . . . . . . . . . . . . . 552.2.11 Canonical Commutation Relations . . . . . . . . . . . . . . . . . . 622.2.12 Matrix Elements of Unitary Operators (Changing Representation) 632.2.13 Representation Independent Properties of Operators . . . . . . . . 672.2.14 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . 692.2.15 Eigenproperties of Hermitian Operators . . . . . . . . . . . . . . . 702.2.16 Obtaining Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . 712.2.17 Common Eigenstates of Commuting Observables . . . . . . . . . . 77
4 CONTENTS
2.3 Postulate III: The Measurement of Quantum Mechanical Systems . . . . . 802.3.1 Sum of Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 852.3.2 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.3.3 Statistical Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 872.3.4 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . 882.3.5 Preparation of a State Using a CSCO . . . . . . . . . . . . . . . . 90
2.4 Postulate IV : Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.4.1 Construction of the Hamiltonian and Other Observables . . . . . . 932.4.2 Some Features of Quantum Mechanical Evolution . . . . . . . . . . 942.4.3 Evolution of Mean Values . . . . . . . . . . . . . . . . . . . . . . . 962.4.4 Eherenfest’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 972.4.5 Evolution of Systems with Time Independent Hamiltonians . . . . 992.4.6 The Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . 102
3 The Harmonic Oscillator 1053.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.1.1 Algebraic Approach to the Quantum Harmonic Oscillator . . . . . 1073.1.2 Spectrum and Eigenstates of the Number Operator N . . . . . . . 1103.1.3 The Energy Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.1.4 Action of Various Operators in the Energy Representation . . . . . 1173.1.5 Time Evolution of the Harmonic Oscillator . . . . . . . . . . . . . 121
4 Bound States of a Central Potential 1234.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2 Hydrogenic Atoms: The Coulomb Problem . . . . . . . . . . . . . . . . . 1274.3 The 3-D Isotropic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 Approximation Methods for Stationary States 1335.1 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.2 Perturbation Theory for Nondegenerate Levels . . . . . . . . . . . . . . . 1375.3 Perturbation Theory for Degenerate States . . . . . . . . . . . . . . . . . 146
5.3.1 Application: Stark Effect of the n = 2 Level of Hydrogen . . . . . 148
6 Many Particle Systems 1516.1 The Direct Product of Linear Vector Spaces . . . . . . . . . . . . . . . . . 151
6.1.1 Motion in 3 Dimensions Treated as a Direct Product of Vector Spaces1556.1.2 The State Space of Spin-1/2 Particles . . . . . . . . . . . . . . . . 156
6.2 The State Space of Many Particle Systems . . . . . . . . . . . . . . . . . . 1566.3 Evolution of Many Particle Systems . . . . . . . . . . . . . . . . . . . . . 1586.4 Systems of Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.4.1 Construction of the Symmetric and Antisymmetric Subspaces . . . 1626.4.2 Number Operators and Occupation Number States . . . . . . . . . 1676.4.3 Evolution and Observables of a System of Identical Particles . . . 1706.4.4 Fock Space as a Direct Sum of Vector Spaces . . . . . . . . . . . . 1756.4.5 The Fock Space of Identical Bosons . . . . . . . . . . . . . . . . . 1776.4.6 The Fock Space of Identical Fermions . . . . . . . . . . . . . . . . 1786.4.7 Observables of a System of Identical Particles Revisited . . . . . . 1826.4.8 Field Operators and Second Quantization . . . . . . . . . . . . . . 186
CONTENTS 5
7 Angular Momentum and Rotations 1897.1 Orbital Angular Momentum of One or More Particles . . . . . . . . . . . 1897.2 Rotation of Physical Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1927.3 Rotations in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 1967.4 Commutation Relations for Scalar and Vector Operators . . . . . . . . . . 1987.5 Relation to Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . 2007.6 Eigenstates and Eigenvalues of Angular Momentum Operators . . . . . . 2027.7 Orthonormalization of Angular Momentum Eigenstates . . . . . . . . . . 2077.8 Orbital Angular Momentum Revisited . . . . . . . . . . . . . . . . . . . . 2117.9 Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.9.1 Irreducible Invariant Subspaces . . . . . . . . . . . . . . . . . . . . 2167.9.2 Rotational Invariance of States . . . . . . . . . . . . . . . . . . . . 2207.9.3 Rotational Invariance of Operators . . . . . . . . . . . . . . . . . . 220
7.10 Addition of Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . 2217.11 Reducible and Irreducible Tensor Operators . . . . . . . . . . . . . . . . . 2307.12 Tensor Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . 2347.13 The Wigner Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 235
8 Time Dependent Perturbations: Transition Theory 2398.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2398.2 Periodic Perturbations: Fermi’s Golden Rule . . . . . . . . . . . . . . . . 2458.3 Perturbations that Turn On . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.3.1 Sudden Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 2518.3.2 The Adiabatic Theorem . . . . . . . . . . . . . . . . . . . . . . . . 252
8.4 Appendix: Landau-Zener Transitions . . . . . . . . . . . . . . . . . . . . . 2548.5 Free Particle Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2598.6 Particle in a time dependent field . . . . . . . . . . . . . . . . . . . . . . . 260
9 Scattering Theory 2639.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.2 An Integral Equation for the Scattering Eigenfunctions . . . . . . . . . . . 268
9.2.1 Evaluation Of The Green’s Function . . . . . . . . . . . . . . . . . 2699.3 The Born Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729.4 Scattering Amplitudes and T-Matrices . . . . . . . . . . . . . . . . . . . . 2729.5 Partial Wave Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
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Chapter 4MANY PARTICLE SYSTEMS
The postulates of quantum mechanics outlined in previous chapters include no restrictionsas to the kind of systems to which they are intended to apply. Thus, although we haveconsidered numerous examples drawn from the quantum mechanics of a single particle,the postulates themselves are intended to apply to all quantum systems, including thosecontaining more than one and possibly very many particles.
Thus, the only real obstacle to our immediate application of the postulates to a systemof many (possibly interacting) particles is that we have till now avoided the questionof what the linear vector space, the state vector, and the operators of a many-particlequantum mechanical system look like. The construction of such a space turns out to befairly straightforward, but it involves the forming a certain kind of methematical productof di¤erent linear vector spaces, referred to as a direct or tensor product. Indeed, thebasic principle underlying the construction of the state spaces of many-particle quantummechanical systems can be succinctly stated as follows:
The state vector jÃi of a system of N particles is an element of the directproduct space
S(N) = S(1) S(2) ¢ ¢ ¢ S(N)formed from the N single-particle spaces associated with each particle.
To understand this principle we need to explore the structure of such direct prod-uct spaces. This exploration forms the focus of the next section, after which we will returnto the subject of many particle quantum mechanical systems.
4.1 The Direct Product of Linear Vector Spaces
Let S1 and S2 be two independent quantum mechanical state spaces, of dimension N1 andN2, respectively (either or both of which may be in…nite). Each space might represent thatof a single particle, or they may be more complicated spaces, each associated with a fewor many particles, but it is assumed that the degrees of mechanical freedom representedby one space are independent of those represented by the other. We distinguish states ineach space by superscripts. Thus, e.g., jÃi(1) represents a state in S1 and jÁi(2) a state ofS2. To describe the combined system we now de…ne a new vector space
S12 = S1 S2 (4.1)
of dimension N12 = N1 £N2 which we refer to as the direct or tensor product of S1 andS2. Some of the elements of S12 are referred to as direct or tensor product states, and are
124 Many Particle Systems
formed as a direct product of states from each space. In other words, from each pair ofstates jÃi(1) 2 S1 and jÁi(2) 2 S2 we can construct an element
jÃ; Ái ´ jÃi(1) jÁi(2) = jÃi(1)jÁi(2) 2 S12 (4.2)
of S12; in which, as we have indicated, a simple juxtaposition of elements de…nes the tensorproduct state when there is no possibility of ambiguous interpretation. By de…nition,then, the state jÃ; Ái represents that state of the combined system in which subsystem1 is de…nitely in state jÃi(1) and subsystem 2 is in state jÁi(2): The linear vector spaceS12 , which is intended to describe the combined system, consists precisely of all suchdirect product states as well as all possible linear combinations of those states. This directproduct of states is assumed to be commutative in the trivial sense that there is nothingspecial about taking the elements in the reverse order, i.e.,
jÃ; Ái ´ jÃi(1)jÁi(2) = jÁi(2)jÃi(1) (4.3)
except that in the abbreviated notation on the left hand side we agree to chose a distinctordering of the spaces once and for all and thus associate the …rst symbol in the list withthat part of the state arising from S1 and the second symbol for that part of the statearising from S2. In the decoupled form on the right, however, we are free to move thetwo kets from each space past each other whenever it is convenient. The tensor productis also assumed to be linearly distributive in the sense that if jÃi(1) = ®j»i(1) + ¯j´i(1),then
jÃ; Ái ´ jÃi(1)jÁi(2) = [®j»i(1) + ¯j´i(1)]jÁi(2) = ®j»; Ái+ ¯j´; Ái; (4.4)
and similarly for kets jÁi(2) which are linear combinations in S2. It is important toemphasize that there are many states in the space S12 that are not direct product states,although (by construction) any state in the product space can be written as a linearcombination of such states. On the other hand, for any given linear combination
jÃi = ®j»; Ái+ ¯j´; Âi (4.5)
of product states, there may or may not be other states in S1 and S2 which allow jÃi to be“factored” into a single direct product of states from each space. If no such factorizationexists, then the state is said to be an entangled state of the combined system. Undersuch circumstances neither subsystem can be described independently by its own statevector, without consideration of the state of the other. Generally, such entanglementsarise as a result of interactions between the component degrees of freedom of each space.The space combined space S12 consists of all possible direct product states as well as allpossible entangled states.
We denote by hÃ; Áj the bra of the dual space S¤12 = S¤1 S¤2 adjoint to the ket jÃ; Ái.Thus, the combined symbol Ã; Á labeling the state is untouched by the adjoint process:
[jÃ; Ái]+ = [jÃi(1)jÁi(2)]+ = [hÃj(1)hÁj(2)] = hÃ; Áj: (4.6)
Inner products taken between elements of the direct product space are obtained bystraightforward linear extension of inner products in each factor space, with the stip-ulation that it is only possible to take inner products between those factors in the samespace, i.e.,
hÃ; Áj´; »i =³hÃj(1)hÁj(2)
´³j´i(1)j»i(2)
´= hÃj´i(1)hÁj»i(2): (4.7)
The Direct Product of Linear Vector Spaces 125
Thus, kets and bras in one space commute past those of the other to form a bracket withmembers of their own space. Since any state is expressible as a linear combination ofproduct states, this completely speci…es the inner product in the combined space.
Basis vectors for the product space S12 can be constructed from basis vectors in thefactor spaces S1 and S2. Speci…cally, if the states fjÁii(1)g form a discrete ONB for S1and the states fjÂji(2)g form a discrete ONB for S2, then the set of N1 £ N2 productstates fjÁi; Âjig form an ONB for the tensor product space S12. We write:
hÁi; Âj jÁi0 ; Âj0i = ±i;i0±j;j0 (4.8)Xi;j
jÁi; ÂjihÁi; Âj j = 1 (4.9)
to denote the orthonormality and completeness of the product basis in S12. Any state inthe system can be expanded in such a basis in the usual way, i.e.,
jÃi =Xi;j
jÁi; ÂjihÁi; Âj jÃi =Xi;j
Ãij jÁi; Âji: (4.10)
Similar relations hold for direct product bases formed form continuous ONB’s in S1 and
S2: Thus, if fj»i(1)g and fjÂi(1)g form continuous ONB’s for the two (in…nite dimensional)factor spaces then the product space is spanned by the basis vectors j»; Âi; for which wecan write
h»; Âj»0; Â0i = ±(» ¡ »0)±(¡ Â0) (4.11)Zd»
Zd j»; Âih»; Âj = 1 (4.12)
jÃi =Zd»
Zd j»; Âih»; ÂjÃi =
Xi;j
Ã(»; Â)j»; Âi: (4.13)
Finally, we can also form direct product bases using a discrete basis for one space and acontinuous basis for the other.Note, that by unitary transformation in the product space it is generally possible toproduce bases which are not the direct products of bases in the factor spaces (i.e., ONB’sformed at least partially from entangled states). Note also, that we have implicitly de…nedoperators in the product space through the last relation.
More generally, operators in S12 are formed from linear combinations of (whatelse) the direct product of operators from each space. That is, for every pair of linearoperators A of S1 and B of S2 we associate an operator AB = A B which acts in S12in such a way that each operator acts only on that part of the product state with whichit is associated. Thus,
ABjÃ;Âi =³AjÃi(1)
´³BjÂi(2)
´= jÃA; ÂBi: (4.14)
Every operator in the individual factor spaces has a natural extension into the productspace, since it can be multiplied by the identity operator of the other space; i.e., theextension of the operator A(1) of S1 into S12 is the operator
A(12) = A(1) 1(2), (4.15)
126 Many Particle Systems
where 1(2) is the identity operator in S2. Often we will drop the superscripts, sincethe symbol A represents the same physical observable in S1 and in S12 (but is generallyunde…ned in S2). Identical constructions hold for the extension of operators of S2. Clearly,1(12) = 1(1)1(2) = 1(1)1(2). Again, as with direct product states, the order of the factorsis not important, so that operators of one factor space always commute with operators ofthe other, while operators from the same space retain the commutation relations thatthey had in the original space This implies, for example, that if Ajai(1) = ajai(1) andBjbi(2) = bjbi(2), then
ABja; bi = BAja; bi =³Ajai(1)
´³Bjbi(2)
´= abja; bi (4.16)
so that the eigenstates of a product of operators from di¤erent spaces are simply productsof the eigenstates of the factors.
As with the states, a general linear operator in S12 can be expressed as a linear combina-tion of direct product operators, but need not be factorizeable into such a product itself.A simple example is the sum or di¤erence of two operators, one from each space; again ifAjai(1) = ajai(1) and Bjbi(2) = bjbi(2), then
(A§B)ja; bi = (a§ b)ja; bi: (4.17)
In general any operator can be expanded in terms of an ONB for the product space inthe usual way, e.g.,
H =Xi;j
Xi0;j0
jÁi; ÂjiHij;i0j0hÁi0 ; Âj0 j; (4.18)
where Hij;i0j0 = hÁi; Âj jHjÁi0 ; Âj0i. Note that if H = AB is a product of operators fromeach space, then the matrix elements representing H in any direct product basis is justthe product of the matrix elements of each operator as de…ned in the factor spaces, i.e.,
hÁi; Âj jHjÁi0 ; Âj0i = hÁi; Âj jABjÁi0 ; Âj0i = hÁijAjÁi0ihÂj jBjÂj0i
The resulting N1 £N2 dimensional matrix in S12 is then said to be the direct or tensorproduct of the matrices representing A in S1 and B in S2:
Finally we note that this de…nition of direct product spaces is easily extensible totreat multiple factor spaces. Thus, e.g., if S1; S2; and S3 are three independent quantummechanical state spaces, then we can take the 3-fold direct product of states jÃi(1) 2 S1and jÁi(2) 2 S2; and jÂi(3) 2 S3 to construct elements jÃ;Á; Âi = jÃi(1)jÁi(2)jÂi(3) of thedirect product space
S123 = S1 S2 S3with inner products
hÃ0; Á0; Â0jÃ; Á; Âi = hÃ0jÃi(1)hÁ0jÁi(2)hÂ0jÂi(2)
and operatorsABCjÃ; Á; Âi = jÃA; ÁB ; ÂCi:
The Direct Product of Linear Vector Spaces 127
4.1.1 Motion in 3 Dimensions Treated as a Direct Product of Vector Spaces
To make these formal de…nitions more concrete we consider a few examples. Consider,e.g., our familiar example of a single spinless quantum particle moving in 3 dimensions.It turns out that this space can be written as the direct product
S3D = Sx Sy Sx: (4.19)
of 3 spaces Si, each of which is isomorphic to the space of a particle moving along onecartesian dimension. In each of the factor spaces we have a basis of position states andrelevant operators, e.g., in Sx we have the basis states, fjxig and operators X;Kx; Px; ¢ ¢ ¢, in Sy the basis states fjyig and operators Y;Ky; Py; ¢ ¢ ¢ , and similarly for Sz. In thedirect product space S3D we can then form, according to the rules outlined in the lastsection, the basis states
j~ri = jx; y; zi = jxi jyi jzi (4.20)
each of which is labeled by the 3 cartesian coordinates of the position vector ~r of R3. Anystate in this space can be expanded in terms of this basis
jÃi =Zdxdy dz jx; y; zihx; y; zjÃi =
Zdxdy dz Ã(x; y; z) jx; y; zi; (4.21)
or in more compact notation
jÃi =Zd3r j~rih~rjÃi =
Zd3r Ã(~r)j~ri: (4.22)
Thus, the state jÃi is represented in this basis by a wave function Ã(~r) = Ã(x; y; z)of 3 variables. Note that by forming the space as the direct product, the individualcomponents of the position operator Xi automatically are presumed to commute withone another, since they derive from di¤erent factor spaces. Indeed, it follows that thecanonical commutation relations
[Xi;Xj ] = 0 = [Pi; Pj ] (4.23)
[Xi; Pj ] = i~±i;j (4.24)
are automatically obeyed due to the rule for extending operators into the product space.The action of the individual components of the position operator and momentum operatorsalso follow from the properties of the direct product space, i.e.,
Xjx; y; zi = Xjxi1yjyi1zjzi = xjx; y; zi (4.25)
PxjÃi = ¡i~Zd3r
@Ã(~r)
@xj~ri (4.26)
and so on.In a similar fashion it is easily veri…ed that all other properties of the space S3D of a singleparticle moving in 3 dimensions follow entirely from the properties of the tensor productof 3 one-dimensional factor spaces. Of course, in this example, it is merely a question ofmathematical convenience whether we view S3 in this way or not.
128 Many Particle Systems
4.1.2 The State Space of Spin-1/2 Particles
Another situation in which the concept of a direct product space becomes valuable is intreating the internal, or spin degrees of freedom of quantum mechanical particles. It is awell-established experimental fact that the quantum state of most fundamental particlesis not completely speci…ed by properties related either to their spatial coordinates or totheir linear momentum. In general, each quantum particle possesses an internal structurecharacterized by a vector observable ~S; the components of which transform under rotationslike the components of angular momentum. The particle is said to possess “spin degreesof freedom”. For the constituents of atoms, i.e., electrons, neutrons, protons, and otherspin-1=2 particles, the internal state of each particle can be represented as a superpositionof two linearly independent eigenvectors of an operator Sz whose eigenvalues s = §1=2(in units of ~) characterize the projection of their spin angular momentum vectors ~S ontosome …xed quantization axis (usually taken by convention to be the z axis). A particlewhose internal state is the eigenstate with s = 1=2 is said to be spin up, with s = ¡1=2;spin down.
The main point of this digression, of course, is that the state space of a spin-1=2particle can be represented as direct product
Sspin-1=2 = Sspatial Sspinof a quantum space Sspatial describing the particle’s spatial state (which is spanned, e.g., byan in…nite set of position eigenstates j~ri), and a two-dimensional quantum space Sspin de-scribing the particle’s internal structure. This internal space is spanned by the eigenstatesjsi of the Cartesian component Sz of its spin observable ~S; with eigenvalues s = §1=2:The direct product of these two sets of basis states from each factor space then generatesthe direct product states
j~r; si = j~ri jsi;which satisfy the obvious orthonormality and completeness relationsX
s=§1=2
Zd3r j~r; sih~r; sj = 1 h~r0; s0j~r; si = ±(~r ¡ ~r0)±s;s0 :
An arbitrary state of a spin-1=2 particle can then be expanded in this basis in the form
jÃi =X
s=§1=2
Zd3r Ãs(~r)j~r; si =
Zd3r [Ã+(~r)j~r; 1=2i+ á(~r)j~r;¡1=2i]
and thus requires a two component wave function (or spinor). In other words, to specifythe state of the system we must provide two seprate complex-valued functions Ã+(~r) andá(~r); with jÃ+(~r)j2 describing the probability density to …nd the particle spin-up at ~rand já(~r)j2 describing the density to …nd the particle spin-down at ~r. Note that, byconstruction, all spin related operators (~S; Sz; S2; etc.) automatically commute with allspatial related operators (~R; ~K; ~P ; etc.).
Thus, the concept of a direct product space arises in many di¤erent situtationsin quantum mechanics and when properly identi…ed as such can help to elucidate thestructure of the underlying state space.
4.2 The State Space of Many Particle Systems
We are now in a position to return to the topic that motivated our interest in directproduct spaces in the …rst place, namely, the construction of quantum states of many
The State Space of Many Particle Systems 129
particle systems. The guiding principle has already been state, i.e., that the state vectorof a system of N particles is an element of the direct product space formed from the Nsingle-particle spaces associated with each particle.
Thus, as the simplest example, consider a collection of N spinless particles each moving inone-dimension, along the x-axis, say (e.g., a set of particles con…ned to a quantum wire).The ®th particle of this system is itself associated with a single particle state space S(®)that is spanned by a set of basis vectors fjx®ig, and is associated with the standard setof operators X®;K®; P®; etc. The combined space S(N) of all N particles in this systemis then the N -fold direct product
S(N) = S(1) S(2) ¢ ¢ ¢ S(N) (4.27)
of the individual single-particle spaces, and so is spanned by the basis vectors formed fromthe position eigenstates of each particle, i.e., we can construct the direct product basis
jx1; : : : ; xNi = jx1i(1) jx2i(2) : : : jxNi(N): (4.28)
In terms of this basis an arbitraryN -particle quantum state of the system can be expandedin the form
jÃi =
Zdx1 : : : dxN jx1; : : : ; xNihx1; : : : ; xN jÃi
=
Zdx1 : : : dxN Ã(x1; : : : ; xN) jx1; : : : ; xNi: (4.29)
Thus, the quantum mechanical description involves a wave function Ã(x1; : : : ; xN) whichis a function of the position coordinates of all particles in the system. This space is clearlyisomorphic to that of a single particle moving in N dimensions, but the interpretationis di¤erent. For a single particle in N dimensions the quantity Ã(x1; : : : ; xN) representsthe amplitude that a position measurement of the particle will …nd it located at thepoint ~r having the associated cartesian coordinates x1; : : : ; xN . For N particles moving inone dimension, the quantity Ã(x1; : : : ; xN) represents the amplitude that a simultaneousposition measurement of all the particles will …nd the …rst at x1, the second at x2, andso on.
The extension to particles moving in higher dimensions is straightforward. Thus, forexample, the state space of N spinless particles moving in 3 dimensions is the tensorproduct of the N single particle spaces S(®) each describing a single particle moving in3 dimensions. The ®th such space is now spanned by a set of basis vectors fj~r®ig, and isassociated with the standard set of vector operators ~R®; ~K®; ~P®; etc. We can now expandan arbitrary state of the combined system
jÃi =
Zd3r1 : : : d
3rN j~r1; : : : ; ~rNih~r1; : : : ; ~rN jÃi
=
Zd3r1 : : : d
3rN Ã(~r1; : : : ; ~rN) j~r1; : : : ; ~rNi: (4.30)
in the direct product basis fj~r1; : : : ; ~rNig of position localized states, each of which de-scribes a distinct con…guration of the N particles. The wave function is then a functionof the N position vectors ~r® of all of the particles (or of the 3N cartesian componentsthereof). A little re‡ection shows that the mathematical description of N particles moving
130 Many Particle Systems
in 3 dimensions is mathematically equivalent, both classically and quantum mechanically,to a single particle moving in space of 3N dimensions.
Before discussing other properties of many-particle systems, it is worth pointing out thatour choice of the position representation in the examples presented above is arbitrary. Thestate of a system of N spinless particles moving in 3 dimensions may also be expanded inthe ONB of momentum eigenstates
jÃi =Zd3k1 : : : d
3kN j~k1; : : : ;~kNih~k1; : : : ;~kN jÃi
=
Zd3k1 : : : d
3kN Ã(~k1; : : : ;~kN) j~k1; : : : ;~kNi;
or in any other complete direct product basis. In addition, it should be noted that therules associated with forming a direct product space ensure that all operators associatedwith a given particle automatically commute with all the operators associated with anyother particle.
4.3 Evolution of Many Particle Systems
The evolution of a many particle quantum system is, as the basic postulates assert, gov-erned through the Schrödinger equation
i~ ddtjÃi = HjÃi (4.31)
where H represents the Hamiltonian operator describing the total energy of the manyparticle system. For a system of N particles, the Hamiltonian can often be written in theform
H =NX®=1
P 2®2m®
+ V (~R1; ~R2; : : : ;RN): (4.32)
As for conservative single particle systems, the evolution of the system is most easilydescribed in terms of the eigenstates of H, i.e., the solutions to the energy eigenvalueequation
HjÁEi = EjÁEi: (4.33)
Projecting this expression onto the position representation leads to a partial di¤erentialequation
NX®=1
¡~22m®
r2®ÁE + V (~r1; ~r2; : : : ; ~rN)ÁE = EÁE (4.34)
for the many particle eigenfunctions ÁE(~r1; ~r2; : : : ; ~rN). When N is greater than twothis equation is (except in special cases) analytically intractable (i.e., nonseparable). Thisanalytical intractability includes the important physical case in which the potential energyof the system arises from pairwise interactions of the form
V =1
2
X®;¯®6=¯
V (~r® ¡ ~r¯): (4.35)
Evolution of Many Particle Systems 131
Solutions of problems of this sort are fundamental to the study of atomic and molecularphysics when N is relatively small (N · 200, typically) and to the study of more gen-eral forms of matter (i.e., condensed phases, liquids, solids, etc.) when N is very large(N » 1024). Under these circumstances one is often led to consider the development ofapproximate solutions developed, e.g., using the techniques of perturbation theory.
An important, and in principle soluble special case is that of noninteracting particles, forwhich the potential can be written in the form
V =NX®=1
V®(~r®) (4.36)
corresponding to a situation in which each particle separately responds to its own externalpotential. In fact, using potentials of this type it is often possible, in an approximate sense,to treat more complicated real interactions such as those described by (4.35). In fact, ifall but one of the particles in the system were …xed in some well-de…ned state, thenthis remaining particle could be treated as moving in the potential …eld generated by allthe others. If suitable potentials V®(~r®) could thus be generated that, in some averagesense, took into account the states that the particles actually end up in, then the actualHamiltonian
H =X®
P 2®2m®
+ V
could be rewritten in the form
H =X®
·P 2®2m®
+ V®(~r®)
¸+¢V
= H0 +¢V
where¢V = V ¡
X®
V®(~r®)
would, it is to be hoped, represent a small perturbation. The exact solution could then beexpanded about the solutions to the noninteracting problem associated with the Hamil-tonian
H0 =X®
·P 2®2m®
+ V®(~r®)
¸:
In this limit, it turns out, the eigenvalue equation can, in principle, be solved by themethod of separation of variables. To obtain the same result, we observe that in this limitthe Hamiltonian can be written as a sum
H0 =NX®=1
·P 2®2m®
+ V®(~R®)
¸=
NX®=1
H®; (4.37)
of single particle operators, where the operator H® acts only on that part of the stateassociated with the single particle space S(®). In each single particle space the eigenstatesjEn®i of H® form an ONB for the associated single particle space. Thus the many particlespace has as an ONB the simultaneous eigenstates jEn1 ; En2 ; : : : ; EnN i of the commut-ing set of operators fH1;H2; : : : ;HNg. These are automatically eigenstates of the total
132 Many Particle Systems
Hamiltonian H0, i.e.,
H0jEn1 ;En2 ; : : : ; En® ; : : : ; EnN i =X®
H®jEn1 ;En2 ; : : : ; En® ; : : : ; EnN i
=X®
En® jEn1 ; En2 ; : : : ; En® ; : : : ; EnN i
= E jEn1 ; En2 ; : : : ; En® ; : : : ; EnN i; (4.38)
where the total energy E =P®En® is just the sum of the single particle energies (as it
is classically). The corresponding wave function associated with such a state is
h~r1; ~r2; : : : ; ~rN jEn1 ; En2 ; : : : ; EnN i = Án1(~r1)Án2(~r2) : : : ÁnN (~rN) (4.39)
which is just the product of the associated single particle eigenfunctions of the operatorsH®, the same result that one would …nd by using the process of separation of variables.Indeed, the standard process of solving a partial di¤erential equation by the method ofseparation of variables can be interpreted as the decomposition of an original functionalspace of several variables into the direct product of the functional spaces associated witheach.
4.4 Systems of Identical Particles
The developments in this chapter derive their importance from the fact that there existexperimental systems of considerable interest which contain more than one particle. It isuseful at this point to consider the implications of another important empirical fact: inmany of these systems the particles of interest belong to distinct classes of (apparently)indistinguishable or identical particles. We use names (protons, electrons, neutrons,silver ions, etc.) to distinguish the di¤erent classes of indistinguishable particles from oneanother. Operationaly, this means that two members of a given class (e.g., two electrons)are not just similar, but are in fact identical to one another, i.e., that there exists noexperiment which could possibly distinguish one from the other. This leads us to ask thefollowing question: What constraint, if any, does indistinguishability impose upon thestate vector of a system of identical particles?
To explore this question, consider …rst a system of N distinguishable, but physicallysimilar, particles each of which is associated with a state space S(®) which is isomorphicto all the others. If the set of vectors fjÁºi(®) j º = 1; 2; ¢ ¢ ¢ g forms an ONB for the spaceof the ®th particle then the many particle space S(N) is spanned by basis vectors of theform
jÁº1 ; Áº2 ; : : : ; ÁºN i = jÁº1i(1)jÁº2i(2) : : : jÁºN i(N) (4.40)
which corresponds to a state in which particle 1 is in state Áº1 ; particle 2 in state Áº2 ;and so on.
Consider, now, the operation of “interchanging” two of the particles in the system.Formally, we can de…ne a set of N(N ¡ 1)=2 unitary exchange operators U®¯ throughtheir action on any direct product basis as follows:
U®¯jÁº1 ; : : : ; Áº® ; : : : ; Áº¯ ; : : : ; ÁºN i = jÁº1 ; : : : ; Áº¯ ; : : : ; Áº® ; : : : ; ÁºN i (4.41)
which puts particle ® in the state formerly occupied by particle ¯, puts particle ¯ inthe state formerly occupied by particle ®, and leaves all other particles alone (we assume
Systems of Identical Particles 133
® 6= ¯). We note in passing that each exchange operator is unitary since it maps anydirect product basis onto itself, but in a di¤erent order. We note also that the productof any exchange operator with itself gives the identity operator, as is easily veri…ed bymultiplying the equation above by U®¯,.and which is consistent with the intuitive ideathat two consecutive exchanges is equivalent to none. Thus, we deduce that U2®¯ = 1:
These properties of the exchange operators aside, the important physical point isthat for distinguishable particles an exchange of this sort leaves the system in a physicallydi¤erent state (assuming º® 6= º¯). The two states jÁº1 ; : : : ; Áº¯ ; : : : ; Áº® ; : : : ; ÁºN i andU®¯jÁº1 ; : : : ; Áº¯ ; : : : ; Áº® ; : : : ; ÁºN i, are linearly independent.
When we mentally repeat this exercise of exchanging particles with a system of indistin-guishable particles, however, we must confront the fact that there can be no experimentwhich can tell which particle is in which state, since there is no way of distinguishing thedi¤erent particles in the system from one another. That is to say, we cannot know thatparticle ® is in the state jÁº®i, but only that the state jÁº®i is occupied by one of theparticles. Thus, an ordered list enumerating which particles are in which states, such asthat labeling the direct product states above, contains more information than is actuallyknowable. For the moment, let jÁº1 ; : : : ; ÁºN i(I) denote the physical state of a systemof N indistinguishable particles in which the states Áº1 ; : : : ; ÁºN are occupied. We theninvoke the principle of indistinguishability and assert that this state and the one
U®¯jÁº1 ; : : : ; ÁºN i(I) (4.42)
obtained from it by switching two of the particles must represent the same physical stateof the system. This means that they can di¤er from one another by at most a phasefactor, i.e., a unimodular complex number of the form ¸ = eiµ. Thus, we assert that thereexists some ¸ for which
U®¯jÁº1 ; : : : ; ÁºN i(I) = ¸jÁº1 ; : : : ; ÁºN i(I): (4.43)
Note, however, that exchanging two particles twice in succession must return the systemto its original state, i.e.,
U®¯[U®¯jÁº1 ; : : : ; ÁºN i(I)] = jÁº1 ; : : : ; ÁºN i(I): (4.44)
This last statement is true whether applied to distinguishable or indistinguishable parti-cles. The implication for the undetermined phase factor, however, is that
U2®¯jÁº1 ; : : : ; ÁºN i(I)] = ¸2jÁº1 ; : : : ; ÁºN i(I) = jÁº1 ; : : : ; ÁºN i(I) (4.45)
from which we deduce that ¸2 = 1. This implies that ¸ = §1, so thatU®¯jÁº1 ; : : : ; ÁºN i(I) = §jÁº1 ; : : : ; ÁºN i(I): (4.46)
Thus, we have essentially proved the following theorem: The physical state of a systemof N identical particles is a simultaneous eigenstate of the set of exchange operatorsfU®¯g with eigenvalue equal either to +1 or ¡1. A state is said to be symmetric underexchange of the particles ® and ¯ if it is an eigenvector of U®¯ with eigenvalue +1;and is antisymmetric if it is an eigenvector of U®¯ with eigenvalue ¡1. It is totallysymmetric if it is symmetric under all exchanges and totally antisymmetric if it isantisymmetric under all exchanges.
Now it is not hard to see that the set of all totally symmetric states of N particlesforms a subspace S(N)S of the original product space S(N), (since any linear combination
134 Many Particle Systems
of such states is still totally symmetric). We shall call S(N)S the symmetric subspaceof S(N). Similarly the set of all totally antisymmetric states forms the antisymmetricsubspace S(N)A of S(N). Our theorem shows that all physical states of a system of N
indistinguishable particles must lie either in S(N)S or S(N)A . (We can’t have a physical statesymmetric under some exchanges and antisymmetric under others, since this would implya physical di¤erence between some of the particles.) Moreover, if a given class of identicalparticles had some states that were symmetric and some that were antisymmetric, itwould be possible to form linear combinations of each, forming physical states that wereneither, and thus violating our theorem. We deduce, therefore, that each class of identicalparticles can have physical states that lie only in S(N)S or only in S(N)A ; it cannot havesome states that are symmetric and some that are antisymmetric.
Experimentally, it is indeed found that the identi…able classes of indistinguishable particlesdivide up naturally into those whose physical states are all antisymmetric and thosewhose physical states are all symmetric under the exchange of any two particles in thesystem. Particles which are antisymmetric under exchange, such as electrons, protons,and neutrons, are referred to as fermions. Particles which are symmetric under exchange,such as photons, are referred to as bosons.
4.4.1 Construction of the Symmetric and Antisymmetric Subspaces
We …nd ourselves in an interesting formal position. We have found that it is a straight-forward exercise to construct the Hilbert space S(N) of N distinguishable particles as adirect product of N single particle spaces. We see now that the physical states associatedwith N indistinguishable particles are necessarily restricted to one or the other of twosubspaces of the originally constructed direct product space. We still have not said howto actually construct these subspaces or indeed how to actually produce a physical stateof such a system. This would be straightforward, of course, if we had at our disposal theprojectors PS and PA onto the corresponding symmetric and antisymmetric subspaces,for then we could start with any state in S(N) and simply project away those parts ofit which were not symmetric or antisymmetric, respectively. These projectors, if we canconstruct them, must satisfy the condition obeyed by any projectors, namely,
P 2S = PS P 2A = PA: (4.47)
In addition, if jÃSi and jÃAi represent, respectively, any completely symmetric or anti-symmetric states in S(N), then we must have
PSjÃSi = jÃSi PAjÃSi = 0 (4.48)
PAjÃAi = jÃAi PSjÃAi = 0 (4.49)
the right-hand relations follow because an antisymmetric state must be orthogonal toall symmetric states and vice versa. It turns out that the projectors PS and PA arestraightforward to construct once one has assembled a rather formidable arsenal of unitaryoperators referred to as permutation operators, which are very closely related to, andin a sense constructed from, the exchange operators. To this end it is useful to enumeratesome of the basic properties of the exchange operators U®¯ :
1. All exchange operators are Hermitian, since, as we have seen, they have real eigen-values ¸ = §1.
2. All exchange operators are nonsingular, since they are equal to their own inverses,a fact that we have already used by observing, e¤ectively, that U2®¯ = 1, henceU¡1®¯ = U®¯.
Systems of Identical Particles 135
3. All exchange operators are unitary since they are Hermitian and equal to their owninverses; it follows that U®¯ = U+®¯ = U
¡1®¯ . Thus each one transforms any complete
direct product basis for S(N) onto another, equivalent, direct product basis.
4. Di¤erent exchange operators do not generally commute. This makes sense on aphysical basis; if we …rst exchange particles ® and ¯ and then exchange particles¯ (which is the original particle ®) and ° we get di¤erent results then if we makethese exchanges in the reverse order.
The product of two or more exchange operators is not, in general, simply another exchangeoperator, but is a unitary operator that has the e¤ect of inducing a more complicatedpermutation of the particles among themselves. Thus, a product of two or more exchangeoperators is one of the N ! possible permutation operators the members of which set wewill denote by the symbol U», where
» ´µ1; 2 ; : : : ; N»1; »2; : : : ; »N
¶(4.50)
denotes an arbitrary permutation, or reordering of the integers (1; 2 ; : : : ; N) into anew order, denoted by (»1; »2; : : : ; »N). Thus the operator U» has the e¤ect of replacingparticle 1 with particle »1, particle 2 with particle »2 and so on, i.e.
U»jÁº1 ; : : : ; ÁºN i = jÁº»1 ; : : : ; Áº»N i
There are N ! such permutations of N particles. The set of N ! permutation operatorsshare the following properties, some of which are given without proof:
1. The product of any two permutation operators is another permutation operator. Infact, they form a group, the identity element of which is the identity permutation thatmaps each particle label onto itself. Symbolically, we can write U»U»0 = U»00 . Thegroup property also insures that this relation applies to the whole set of permutationoperators, i.e., for any …xed permutation operator U» the set of products fU»U»0g isequivalent to the set fU»g of permutation operators itself. This important propertywill be used below.
2. Each permutation », or permutation operator U» can be classi…ed as being either“even” or “odd”. An even permutation operator can be written as a product
U» = U®¯U°± ¢ ¢ ¢U¹ºof an even number of exchange operators, and odd permutation as a product of anodd number of exchange operators. This factorization of each U» is not unique, sincewe can obviously insert an even number of factors of U®¯ in the product withoutchanging the permutation (or the even-odd classi…cation ). This is equivalent to theobservation that an arbitrary permutation of N particles can be built up through aseries of simple exchanges in many di¤erent ways.
3. The exchange parity "» of a given permutation operator is de…ned to be +1 if U»is even and ¡1 if it is odd. Equivalently, if U» can be written as a product of nexchange operators then "» = (¡1)n:
136 Many Particle Systems
It turns out that for a given number N of particles, there are an equal number of evenand odd permutation operators (indeed multiplying any even permutation operator byone exchange operator makes it an odd permutation operator and vice versa.). It is alsofairly easy to see that if a state jÃSi is symmetric (i.e., invariant) under all exchanges, itis also symmetric under any product thereof, and thus is symmetric under the entire setof permutation operators, i.e.,
U»jÃSi = jÃSi: (4.51)
Physically, this means that it is invariant under an arbitrary permutation of the particlesin the system. Alternatively, it means that jÃSi is an eigenstate of U» with eigenvalue 1:
On the other hand, a state jÃAi which is antisymmetric under all exchanges will be leftunchanged after an even number of exchanges (i.e., after being operated on by an evennumber of exchange operators), but will be transformed into its negative under an oddnumber of exchanges. Thus is succinctly expressed by the relation
U»jÃAi = "»jÃAi: (4.52)
Thus, an antisymmetric state is an eigenstate of U» with eigenvalue "».
With these properties in hand, we are now ready to display the form of the projectors ontothe symmetric and antisymmetric subspaces of S(N). These are expressible as relativelysimple sums of the permutation operators as follows; the symmetric projector is essentiallythe symmetric sum
PS =1
N !
X»
U» (4.53)
of all the permutation operators, while the antisymmetric projector has a form
PA =1
N !
X»
U»"»: (4.54)
that weights each of the permutation operators by its exchange parity of §1. Thus, halfthe terms in the antisymmetric projector have a ¡1 and half have a +1.
The proof that these operators do indeed satisfy the basic properties of the pro-jectors as we described earlier is straightforward. First we note that if jÃSi is a symmetricstate, then
PSjÃSi =1
N !
X»
U»jÃSi =1
N !
X»
jÃSi = jÃSi (4.55)
where the sum over the N ! permutations » eliminates the normalization factor. Similarly,we have
PAjÃSi =1
N !
X»
"»U»jÃSi =1
N !
X»
"»jÃSi = 0; (4.56)
where we have used the fact that there are equal numbers of even and odd permutations toevaluate the alternating sum in the last expression. Similarly, if jÃAi is an antisymmetricstate,
PAjÃAi =1
N !
X»
"»U»jÃAi =1
N !
X»
"2» jÃAi = jÃAi (4.57)
whilePS jÃAi =
1
N !
X»
U»jÃAi =1
N !
X»
"»jÃAi = 0: (4.58)
Systems of Identical Particles 137
Finally, we must show that P 2S = PS, and P2A = PA. To show this we note …rst that
U»PS =1
N !
X»0U»U»0 =
1
N !
X»00U»00 = PS; (4.59)
where we have used the group property which ensures that the set of permutation opera-tors simply reproduces itself when multiplied by any single permutation operator. Thusfor the symmetric projector we have
PSPS =1
N !
X»
U»PS =1
N !
X»00PS = PS; (4.60)
showing that it is indeed a projection operator.To prove a similar result for the antisymmetric projector, we note …rst that the
exchange parity of the product of any two permutation operators is the product of theirindividual parities. Thus if U»U»0 = U»00 ; then
"»00 = "»"»0 : (4.61)
This can be seen by factorizing both permutation operators in the product into exchangeoperators. If U» and U»0 contain n and m factors, respectively, then U»00 contains n+mfactors, so "»00 = (¡1)n+m = (¡1)n(¡1)m = "»"»0 . Using this, along with the obviousrelation "2» = 1; it follows that
U»PA =1
N !
X»0U»U»0"»0 =
1
N !
X»0"»U»U»0"»"»0 = "»
1
N !
X»00"»00U»00
= "»PA; (4.62)
Thus for the antisymmetric projector we have
PAPA =1
N !
X»
"»U»PA =1
N !
X»
"2»PA =1
N !
X»
PA = PA; (4.63)
which is the desired idempotency relation for the antisymmetic projector.
Using these projectors, the physical state of a system ofN identical particles is constructedby projection. To each state jÃi 2 S(N) there corresponds at most one state jÃSi 2 S(N)S
and one state jÃAi 2 S(N)A : For bosons, the normalized symmetrical state is given by theprojection
jÃSi =PSjÃiphÃjPSjÃi (4.64)
onto S(N)S ; while for fermions we have
jÃAi =PAjÃiphÃjPAjÃi : (4.65)
It is important to point out that the projection may give the null vector if, for example,the original state is entirely symmetric or antisymmetric to begin with.Example: Two Identical Bosons - Let jÁi and jÂi be two orthonormal single particlestates. The state jÁ(1); Â(2)i = jÁ;Âi = jÃi is in the two-particle direct product spaceS(2): The projection of jÃi onto the symmetric subspace of S(2) is
PSjÃi = 1
2[U12 + U21] jÁ; Âi = 1
2[jÁ; Âi+ jÂ; Ái] : (4.66)
138 Many Particle Systems
To normalize we evaluate
1
2[hÁ; Âj+ hÂ; Áj] 1
2[jÁ; Âi+ jÂ; Ái] = 1
4[1 + 0 + 0 + 1] =
1
2; (4.67)
sojÃSi =
1p2[jÁ; Âi+ jÂ; Ái] : (4.68)
Notice that if Á = Â; then the original state is already symmetric, i.e., PSjÁ; Ái = jÁ; Ái.Thus, it is possible for two (or more) bosons to be in the same single particle state. Alsonotice that if Á 6= Â; then the states jÁ; Âi and jÂ; Ái are both projected onto the samephysical state, i.e., PSjÁ; Âi = PSjÂ; Ái. This fact, namely that there are generally manystates in S(N) that correspond to the same physical state in S(N)S is referred to as a liftingof the exchange degeneracy of S(N).Example: Two Identical Fermions - Again let jÁi and jÂi be two orthonormal singleparticle states, and jÁ(1); Â(2)i = jÁ; Âi = jÃi be the associated two-particle state in S(2):The projection of jÃi onto the antisymmetric part of S(2) is
PAjÃi = 1
2["12U12 + "21U21] jÁ; Âi = 1
2[jÁ; Âi ¡ jÂ; Ái] : (4.69)
To normalize we evaluate
1
2[hÁ; Âj ¡ hÂ; Áj] 1
2[jÁ; Âi ¡ jÂ; Ái] = 1
4[1¡ 0¡ 0 + 1] = 1
2; (4.70)
sojÃAi =
1p2[jÁ; Âi ¡ jÂ; Ái] : (4.71)
Again, the projection of the states jÁ; Âi and jÂ; Ái onto S(2)A correspond to the samephysical state, (a lifting of the exchange degeneracy) although they di¤er from one anotherby a phase factor, i.e., PSjÁ; Âi = ¡PAjÂ;Ái = ei¼PS jÁ; Âi. Notice, however, that ifÁ = Â; then the projection of the original state vanishes. Thus, two identical fermionscannot occupy the same physical state. This fact, which follows from the symmetrizationrequirement is referred to as the Pauli exclusion principle.
It is possible to write the fermion state derived above in a convenient mathemat-ical form involving a determinant, i.e., if we write
jÃAi =1p2
¯jÁ(1)i jÁ(2)ijÂ(1)i jÂ(2)i
¯(4.72)
and formally evaluate the determinant of this odd matrix we obtain
jÃAi =1p2
hjÁ(1)ijÂ(2)i ¡ jÂ(1)ijÁ(2)i
i=
1p2[jÁ; Âi ¡ jÂ; Ái] (4.73)
This determinantal way of expressing the state vector (or the wave function) is referredto as a Slater determinant, and generalizes to a system of N particles. Thus, ifjÁº1i; jÁº2i; ¢ ¢ ¢ ; jÁºN i represent a set of N orthonormal single particle states, then theSlater determinant
jÃAi =1pN !
¯¯ jÁ
(1)º1 i jÁ(2)º1 i ¢ ¢ ¢ jÁ(N)º1 i
jÁ(1)º2 i jÁ(2)º2 i ¢ ¢ ¢ jÁ(N)º2 i¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢jÁ(1)ºN i jÁ(2)ºN i ¢ ¢ ¢ jÁ(N)ºN i
¯¯ (4.74)
Systems of Identical Particles 139
is a properly normalized state of N fermions. Note that if any two states jÁº®i and jÁº¯ iare the same, then the corresponding rows will be identical and the resulting state willvanish, automatically satisfying the Pauli exclusion principle.
4.4.2 Number Operators and Occupation Number States
We now turn to the task of identifying and constructing ONB’s for the symmetric (orbosonic) subspace describing a collection of N identical bosons and the antisymmetric (orfermionic) subspace describing a collection of N identical fermions. As we have seen, froma given set of single particle states fjÁºi jº = 1; 2; ¢ ¢ ¢ g we can form for the direct productspace S(N) of N distinguishable particles an ONB of direct product states jÁº1 ; : : : ; ÁºN i,in which particle 1 is in state jÁº1i; particle 2 in state jÁº2i; and so on. It is useful at thispoint to introduce a set of operators fNº jº = 1; 2; ¢ ¢ ¢ g associated with this representa-tion which “count” the number of particles that are in each single-particle state, i.e., byde…nition,
N1jÁº1 ; : : : ; ÁºN i = n1jÁº1 ; : : : ; ÁºN iN2jÁº1 ; : : : ; ÁºN i = n2jÁº1 ; : : : ; ÁºN i
...
Nº jÁº1 ; : : : ; ÁºN i = nº jÁº1 ; : : : ; ÁºN i... (4.75)
where, e.g., n1 represents the number of times the symbol Á1 appears in the list (Áº1 ; : : : ; ÁºN );i.e., it describes the number of particles in the collection occupying the single-particle statejÁ1i: Similarly, the eigenvalue
nº =NX®=1
±º;º® (4.76)
is, for this state, the number of particles in the single-particle state jÁºi: Thus the directproduct states of this representation are simultaneous eigenstates of this set of numberoperators fNºg ; and each such state is characterized by a speci…c set of eigenvaluesfnºg : The eigenvalues nº are referred to as the occupation numbers associated withthis representation of single particle states. Now, typically, the number of single-particlestates fjÁºig is in…nite, and so most of the occupation numbers fnºg characterizing anygiven basis state jÁº1 ; : : : ; ÁºN i are equal to zero. In fact, it is clear that at mostN of themare not zero, and this maximum can be obtained only if all the particles are in di¤erentsingle-particle states. On the other hand, at least one of the occupation numbers is notzero, since the sum of the occupation numbers is equal to the total number of particles inthe system, i.e.,
N =Xº
nº : (4.77)
Now for distinguishable particles, the occupation numbers fnºg = fn1; n2; ¢ ¢ ¢ ggenerally do not determine uniquely the corresponding basis states. This is because theoccupation numbers, by construction, contain information about which single-particlestates are …lled, but contain no information about which particles are in which states.Thus, e.g., if jÁº1 ; : : : ; ÁºN i is a direct product state characterized by a certain set ofoccupation numbers fnºg ; any permutation of the particles in the system among thesame speci…ed set of single-particle states will leave the system in a state U»jÁº1 ; : : : ; ÁºN ihaving exactly the same set of single-particle states …lled (albeit by di¤erent particlesthan in the original). The set of occupation numbers for any such state will, therefore, beidentical with that of the unpermuted state. If all of the occupied single-particle states
140 Many Particle Systems
Áº1 ; : : : ; ÁºN are distinct (i.e., if all the associated occupation numbers are either zeroor one) , then there will be N ! linearly independent states U»jÁº1 ; : : : ; ÁºN i associatedwith the same set of occupation numbers fnºg. If, however, some of the single-particlestates are multiply occupied, then any permutation that simply rearranges those particlesin the same occupied state will leave the system in exactly the same state as before.If, e.g., there are n1 particles in the state jÁ1i then there will be n1! permutations thatsimply permute these n1 particles among themselves, and so on. Arguing in this way foreach multiply-occupied state, we deduce that the number of linearly-independent states©U»jÁº1 ; : : : ; ÁºN i
ªof N distinguishable particles associated with a given set fnºg of
occupation numbers is given by the expression
g (fnºg) = N !
n1!n2! ¢ ¢ ¢ : (4.78)
Recall that 0! = 1! = 1; so that this expression reduces to N ! when there are N distinctsingle-particle states …lled. This number g (fnºg) de…nes more closely the term exchangedegeneracy introduced earlier, since it is, in fact, the degeneracy in S(N) associated withthe set of simultaneous eigenvalues fnºg of the number operators fNºg, a degeneracy thatarises entirely due to the distinguishability of the particles described.
Now when the basis vectors©jÁº1 ; : : : ; ÁºN iª of this representation of S(N) are
projected onto the symmetric or anti-symmetric subspaces, they generate an ONB foreach of these two smaller subspaces. As we might expect, however, there is a reduc-tion in the number of linearly-independent basis vectors that survive the projection. Inparticular, projection onto either the symmetric or anti-symmetric spaces eliminates anyinformation regarding which particle is in which single-particle state. As a consequence,all the basis vectors
©U»jÁº1 ; : : : ; ÁºN i
ªassociated with a given set fnºg of occupation
numbers project onto at most one linearly independent basis vector of each subspace.This dramatic reduction (which is the essence of the removal of the exchange degeneracythat we observed in the two-particle case) allows us to label each such basis vector bythe associated set of occupation numbers, and provides us with what is referred to asthe occupation number representation associated with a given set of single-particlestates. In what follows we describe the projection process independently for the bosonicand fermionic subspaces.
The projection of the basis vector jÁº1 ; : : : ; ÁºN i onto the symmetric subspaceS(N)S of N -identical bosons is, by de…nition, the vector PSjÁº1 ; : : : ; ÁºN i: On the otherhand the projection of the state U»jÁº1 ; : : : ; ÁºN i; which is associated with the same setof occupation numbers, is given by the vector PSU»jÁº1 ; : : : ; ÁºN i: But, we note that
PSU» =1
N !
X»0U»0U» =
1
N !
X»00U»00 = PS; (4.79)
where we have used the group properties of the permutation operators. Thus we …nd that
PSU»jÁº1 ; : : : ; ÁºN i = PSjÁº1 ; : : : ; ÁºN i; (4.80)
and hence deduce that all g (fnºg) basis vectors of S(N) associated with the same set fnºgof occupation numbers project onto precisely the same vector in S(N)S : As a consequence,the basis vectors obtained in this way by projection onto S(N)S can be uniquely labeled bythe occupation numbers fnºg that characterize them. We thus denote by
jn1; n2; : : :i =PSjÁº1 ; : : : ; ÁºN ijjPSjÁº1 ; : : : ; ÁºN ijj
(4.81)
Systems of Identical Particles 141
the symmetric state of N identical bosons containing n1 particles in state Á1; n2 particlesin state Á2; and so on. The set of such states with
Pº nº = N span the symmetric
subspace S(N)S of N identical bosons, and form what is referred to as the occupationnumber representation associated with this set of single particle states. (Note that anysuch ONB of single particle states generates a similar representation.) For any suchrepresentation for the symmetric subspace, therefore, we can write an orthonormalityrelation
hn1; n2; : : : jn01; n02; : : :i = ±n1;n01±n2;n02 ¢ ¢ ¢ (4.82)
showing that two occupation number states are orthogonal unless they have exactly thesame set of occupation numbers, and a completeness relationX
fnºgjn1; n2; : : :ihn1; n2; : : : j = 1 (4.83)
for the symmetric space, where the sum is over all sets of occupation numbers consistentwith the constraint
Pº nº = N: We note in passing that the number operators fNºg
form a complete set of commuting observables (CSCO) for this symmetric subspace, sinceeach basis vector in this occupation number representation is uniquely labeled by theassociated set of eigenvalues.
Construction of the occupation number representation for the antisymmetricspace of N identical fermions is similar, but some important di¤erences arise. In par-ticular, we note …rst that the projection onto S(N)A of any basis vector jÁº1 ; : : : ; ÁºN i ofS(N)S having more than one particle in any given single particle state vanishes, since thecorresponding Slater determinant (4.74) will have repeated rows. Thus most of the basisstates of S(N) have no physical counterpart in the antisymmetric subspace. (In this sense,threfore, fermionic spaces are always smaller than bosonic spaces.) In general, only thosedirect product states jÁº1 ; : : : ; ÁºN i with each particle in a distinct single-particle stateÁº1 ; : : : ; ÁºN will have a non-vanishing projection PAjÁº1 ; : : : ; ÁºN i onto S(N)A : In thatcase, the projection of the N ! linearly independent states U»jÁº1 ; : : : ; ÁºN i characterizedby the same set fnºg of occupation numbers (which will now all be 0’s or 1’s) will takethe form PAP»jÁº1 ; : : : ; ÁºN i: But this product of operators can also be simpli…ed, i.e.,
PAU» =1
N !
X»0"»0U»0U» =
"»N !
X»0"»0"»U»0U»
= "»
24 1N !
X»00"»00U»00
35 = "»PA (4.84)
where we have used the group properties of the permutation operators and the identity"2» = 1: Thus, we …nd that
PAU»jÁº1 ; : : : ; ÁºN i = "»PAjÁº1 ; : : : ; ÁºN i = §PAjÁº1 ; : : : ; ÁºN i: (4.85)
Thus, the even permutations project onto the vector PAjÁº1 ; : : : ; ÁºN i and the odd per-mutations onto its negative ¡PAjÁº1 ; : : : ; ÁºN i: Of course, although these two states di¤erby a phase factor of unit modulus (¡1 = ei¼), they represent precisely the same physicalstate in Hilbert space. Thus, with a suitable phase convention, we …nd that all of the basisvectors associated with a given acceptable set fnºgof occupation numbers project ontothe same basis vector of S(N)A ; and so are uniquely labeled by the occupation numbers that
142 Many Particle Systems
characterize them. Thus, the number operators fNºg form a complete set of commutingobservables (CSCO) for the antisymmetric subspace, as well. We thus denote by
jn1; n2; : : :i =PAjÁº1 ; : : : ; ÁºN ijjPAjÁº1 ; : : : ; ÁºN ijj
(4.86)
the antisymmetric state of N identical fermions containing n1 particles in state Á1; n2particles in state Á2; and so on, where all nº 2 f0; 1g ; and where Pº nº = N: Tounambiguously …x the phase of the associated occupation number basis state jn1; n2; : : :i;we note that of all the N ! states
©U»jÁº1 ; : : : ; ÁºN i
ªassociated with the same set of
occupation numbers fnºg ; only one of them has the occupied states ordered so that the…rst particle is in the lowest occupied state, the second is in the next-to-lowest occupiedstate, and so on. It is this state (or any even permutation thereof) that we projectonto S(N)A to produce the basis state. Thus jn1; n2; : : :i is identi…ed with the normalizedantisymmetric projection of that state jÁº1 ; : : : ; ÁºN i that has the correct set of occupationnumbers and for which
º1 < º2 < ¢ ¢ ¢ < ºN : (4.87)
Another way of putting it is that we identify the basis state jn1; n2; : : :i with that Slaterdeterminant (4.74) in which the indices º® of the single particle states are strictly increas-ing in going from the top row to the bottom row.
For this set of states, we can write completeness and orthonormality relationsessentially identical to those that we wrote for the occupation number representation ofthe bosonic subspace, except for the restriction on the allowed set of occupation numbersto those for which nº 2 f0; 1g :4.4.3 Evolution and Observables of a System of Identical Particles
The state vector jÃ(t)i describing a collection of identical bosons or fermions, if it isto continue to describe such a system for all times, must remain within the bosonic orfermionic space in which it starts. What does this imply about the structure of thecorresponding Hamiltonian and of the observables for such a system? To address thisquestion we note that, as for any quantum system, evolution of the state vector is governedby Schrödinger’s equation
i~ ddtjÃ(t)i = HjÃ(t)i (4.88)
where H is the Hamiltonian governing the many-particle system. Now the state vectorof a collection of bosons must remain symmetric under the exchange of any two particlesin the system, and the state vector of a collection of identical fermions must remainantisymmetric under any such exchange. Thus, we can write that, for all times t;
U®¯jÃ(t)i = ¸jÃ(t)i (4.89)
where ¸ = +1 for bosons and ¡1 for fermions. Applying U®¯ to the evolution equationwe determine that
i~¸ ddtjÃ(t)i = U®¯HjÃ(t)i
= ¸U®¯HU+®¯jÃ(t)i (4.90)
where we have inserted a factor of 1 = U+®¯U®¯ betweenH and the state vector. Cancelingthe common factor of ¸; and comparing the result to the original evolution equation we…nd that
i~ ddtjÃ(t)i = U®¯HU+®¯jÃ(t)i = HjÃ(t)i (4.91)
Systems of Identical Particles 143
which is satis…ed provided thatH = U®¯HU
+®¯: (4.92)
This shows that H, if it is to preserve the exchange symmetry of the state vector, must beinvariant under the unitary transformations that exchange particles in the system. An op-erator A is said to be symmetric under the exchange of particles ® and ¯ if U®¯AU+®¯ = Aand is said to be antisymmetric under exchange if U®¯AU+®¯ = ¡A. Using this termi-nology, we see that the Hamiltonian of a collection of identical bosons or fermions mustbe symmetric under all particle exchanges in the system. Another way of expressing thesame thing is obtained by multiplying (4.92) through on the right by U®¯ to obtain theresult HU®¯ = U®¯H; or,
[U®¯;H] = 0 (4.93)
Thus, any operator that is symmetric under particle exchange commutes with the ex-change operators. If H commutes with all of the exchange operators, then it commuteswith any product of exchange operators, i.e., with any permutation operator U»; thus
[U»;H] = 0: (4.94)
Since the projectors PS and PA onto the bosonic and fermionic subspaces are linearcombinations of permutation operators, they must also commute with H, i.e.,
[PS;H] = 0 = [PA;H] : (4.95)
From the basic theorems that we derived for commuting observables, it follows that theeigenspaces of PS and PA; in particular, the subspaces S
(N)S and S(N)A ; must be globally-
invariant under the action of the Hamiltonian H. In other words, H connects no stateinside either subspace to any state lying outside the subspace in which it is contained.Indeed, this is the way that H keeps the state vector inside the relevant subspace. Itfollows, for example, that if the symmetric and antisymmetric projectors commute withthe Hamiltonian at each instant, then they also commute with the evolution operatorU(t; t0) which can be expressed as a function of the Hamiltonian. This means, e.g., thatif jÃ(t0)i is some arbitrary initial state vector lying in S(N) which evolves into the statejÃ(t)i = U(t; t0)jÃ(t0)i; then the projection of jÃ(t)i onto the symmetric subspace can bewritten
jÃS(t)i = PSjÃ(t)i = PSU(t; t0)jÃ(t0)i = U(t; t0)PSjÃ(t0)i = U(t; t0)jÃS(t0)i (4.96)
and the projection onto the antisymmetric space can be written
jÃA(t)i = PAjÃ(t)i = PAU(t; t0)jÃ(t0)i = U(t; t0)PAjÃ(t0)i = U(t; t0)jÃA(t0)i (4.97)
which shows that we can project an arbitrary initial state …rst to get an initial statevector with the right symmetry, and then evolve within the subspace, or simply evolvethe arbitrary initial state and project at the end of the evolution process to get the statethat evolves out of the appropriate projection of the initial state.
The fact that the Hamiltonian commutes with PS and PA also means that thereexists an orthonormal basis of energy eigenstates spanning each of the two subspacesof interest. This also makes sense from the point of view of the measurement process.If this were not the case, then an arbitrary state in either subspace would have to berepresented as a linear combination of energy eigenstates some of which lie outside (orhave components that lie outside) the subspace of interest. An energy measurement wouldthen have a nonzero probability of collapsing the system onto one of these inadmissible
144 Many Particle Systems
energy eigenstates, i.e., onto a state that is not physically capable of describing a collectionof identical particles, since it lies outside the relevant subspace.
Clearly, arguments of this sort based upon what can happen during a measure-ment process must apply as well to any observable of a system of identical particles. Thatis, in order for a measurement of an observable A not leave the system in an inadmissiblestate, it must have a complete set of eigenstates spanning the relevant subspace, and must,therefore, be symmetric under all particle exchanges.
So what kind of operators are, in fact, symmetric under all particle exchanges?As we have seen, the number operators fNºg associated with a given direct productrepresentation of S(N) have the property that they count the number of particles in anygiven single particle state, but are completely insensitive to which particles are in whichstate. Indeed, if the state jÁº1 ; : : : ; ÁºN i is characterized by a given set of occupationnumbers fnºg (i.e., is an eigenstate of the number operators fNºg with a particular set ofeigenvalues) then so is any state U»jÁº1 ; : : : ; ÁºN i obtained from this one by a permutationof the particles. In other words, it follows that if
Nº jÁº1 ; : : : ; ÁºN i = nº jÁº1 ; : : : ; ÁºN i (4.98)
then
Nº£U»jÁº1 ; : : : ; ÁºN i
¤= nº
£U»jÁº1 ; : : : ; ÁºN i
¤= U»Nº jÁº1 ; : : : ; ÁºN i: (4.99)
Since this holds for each vector of the basis set, it follows that NºU» = U»Nº ; hence
[U»; Nº ] = 0; (4.100)
which implies that[PS; Nº ] = 0 = [PA; Nº ] : (4.101)
Indeed, the number operators form a CSCO for the two subspaces of interest, and clearlyhave a complete ONB of eigenvectors (i.e., the occupation number states) spanning eachsubspace. It follows that any observable that can be expressed as a function of the numberoperators will also be symmetric under all particle exchanges.
It is also possible to form suitable observables as appropriate linear combinationsof single-particle operators. To see how this comes about, consider as a speci…c example,a system of two particles, each of which we can associate with a position operator ~R1 and~R2 whose eigenstates in the two particle space S2 are the direct product position statesfj~r1; ~r2ig and have the property that
~R1j~r1; ~r2i = ~r1j~r1; ~r2i ~R2j~r1; ~r2i = ~r2j~r1; ~r2i (4.102)
To see what happens under particle exchange, consider the operator
U21 ~R1U21 = U21 ~R1U+21; (4.103)
where U21 = U+21 exchanges particles 1 and 2. Thus, we …nd that
U21 ~R1U21j~r1; ~r2i = U21 ~R1j~r2; ~r1i = U21~r2j~r2; ~r1i= ~r2j~r1; ~r2i = ~R2j~r1; ~r2i (4.104)
Thus, as we might have anticipated,
U21 ~R1U+21 =
~R2: (4.105)
Systems of Identical Particles 145
More generally, in a system of N particles if B® is a single-particle operator associatedwith particle ®; then its transform under U®¯
U®¯B®U+®¯ = B¯ (4.106)
is the corresponding operator for particle ¯. In this light, any symmetric combination ofoperators such as ~R®+ ~R¯ is readily veri…ed to be symmetric under the associated particleexchange, i.e.,
U®¯³~R® + ~R¯
´U+®¯ =
~R¯ + ~R® (4.107)
whereas any antisymmetric combination, such as that related to the relative displacementof two particles is antisymmetric, since
U®¯
³~R® ¡ ~R¯
´U+®¯ =
~R¯ ¡ ~R® = ¡³~R® ¡ ~R¯
´: (4.108)
On the other hand, the relative distance between the particles¯~R¯ ¡ ~R®
¯, being an even
function of the ~R¯ ¡ ~R® is symmetric. Thus any function of a symmetric operator issymmetric, while an even function of antisymmetric operators is also symmetric.
Extending this to a system of N particles, any observable of a system of identicalparticles must be symmetric under all particle exchanges and permutations. This wouldinclude, e.g., any complete sum
B =NX®=1
B® (4.109)
of the corresponding single particle operators for each particle in the system. Examplesof operators of this type include the location of the center of mass
~R =1
N
NX®=1
~R® (4.110)
(recall that all the particles are assumed identical, so the masses are all the same), andthe total linear and angular momentum
~P =NX®=1
~P® ~L =NX®=1
~L®: (4.111)
As we have seen, a system of noninteracting particles has a Hamiltonian that isof precisely this type, i.e.,
H =NX®=1
H® (4.112)
where, e.g.,
H® =P 2®2m
+ V (R®): (4.113)
Actually, although we have constructed these operators as symmetric linear com-binations of single-particle operators, it is easy to show that any operator of this type canalso represented in terms of the number operators associated with a particular occupationnumber representation. Suppose, e.g., that the single-particle states jÁºi(®) are eigenstatesof the single-particle operator B® with eigenvalues bº : Thus, e.g., B®jÁºi(®) = bº jÁºi(®):
146 Many Particle Systems
Then the direct product states formed from this set will be simultaneous eigenstates ofthe all the related operators B®;
B®jÁº1 ; : : : ; Áº® ; : : : ; ÁºN i = bº® jÁº1 ; : : : ; Áº® ; : : : ; ÁºN i (4.114)
and so will be an eigenstate of the symmetric operator B =P®B®; i.e.,
BjÁº1 ; : : : ; ÁºN i =X®
B®jÁº1 ; : : : ; ÁºN i =X®
bº® jÁº1 ; : : : ; Áº® ; : : : ; ÁºN i
= bjÁº1 ; : : : ; Áº® ; : : : ; ÁºN i (4.115)
where the collective eigenvalue b is the sum of the individual single-particle eigenvalues
b =NX®=1
bº® =NX®=1
Xº
bº±º;º® =Xº
bºnº (4.116)
where we have reexpressed the sum over the particle index ® by a sum over the stateindex º; and collected all the eigenvalues associated with the nº particles in the samesingle-particle state Áº together. Thus we can reexpress the eigenvalue equation above as
BjÁº1 ; : : : ; ÁºN i =Xº
bºnº jÁº1 ; : : : ; ÁºN i
=Xº
bºNº jÁº1 ; : : : ; ÁºN i (4.117)
and, through a simple process of identi…cation, reexpress the operator B in the form
B =Xº
bºNº : (4.118)
Thus, e.g., the noninteracting particle Hamiltonian above can be expressed in terms ofthe single particle energy eigenstates jÁºi obeying the equation H®jÁºi = "º jÁºi in theform
H =Xº
"ºNº : (4.119)
There is a real sense in which this way of expressing the Hamiltonian is to be preferred overthe earlier form, particularly when we are dealing with identical particles. The operatorsNº are not labeled by particle numbers, but by the single-particle states that can beoccupied or not. Thus in expressing the Hamiltonian in this fashion, we are not using anotation that suggests (erroneously!) that we can actually label the individual particlesin the system, unlike, e.g., (4.112), which explicitly includes particle labels as well as thenumber of particles in the system. The operator (4.119), on the other hand , makes noexplicit reference either to particle labels or to the number of particles in the system, andtherefore has exactly the same form in any of the spaces S(N) associated with any numberN = 1; 2; ¢ ¢ ¢ of identical particles.
Operators of this type, which can be expressed as a sum of single-particle opera-tors, or equivalently, as a simple linear function of the number operators associated witha particular occupation number representation, we will refer to as one-body operators,since they really depend only on single-particle properties. In addition to these, there arealso operators that depend upon multiple-particle properties. For example, interactionsbetween particles are often represented by “two-body” operators of the form
Vint =1
2
X®;¯® 6=¯
V (~R® ¡ ~R¯) (4.120)
Systems of Identical Particles 147
which is a symmetrized sum of operators that each depend upon the properties of justtwo particles. It turns out that operators of this type can usually be expressed as a simplebilinear function of the number operators associated with some set of single-particle states.Typically, however, the Hamiltonian contains both one-body and two-body parts, and therepresentation in which the one-body part is expressible in the form (4.119) is not onein which the two-body part is expressible as a simple function of the number operators.Conversely, a representation in which the interactions are expressible in terms of numberoperators is not one in which the one-body part is, also. The reason for this is that,typically, the interactions induce transitions between single-particle eigenstates, i.e., theytake particles out of the single-particle states that they occupy and place them backinto other single particle states. In the process, they change the occupation numberscharacterizing the state of the system. It is useful, therefore, to de…ne operators that arecapable of inducing transitions of this type.
We have, in a sense, already encountered operators that do this sort of thing inour study of the harmonic oscillator. The energy eigenstates fjnigof the 1D harmonicoscillator are each characterized by an integer n 2 f0; 1; 2; ¢ ¢ ¢ g that can be viewed as an“occupation number” characterizing the number of vibrational quanta (now considered asa kind of “particle”) in the system. The annihilation, creation, and number operators a;a+; and N = a+a decrease, increase, and count the number of these quanta.
In the 3D harmonic isotropic oscillator, which is separable in Cartesian coordi-nates, the energy eigenstates fjnx; ny; nzig are characterized by a set of three occupationnumbers fnº jº = 1; 2; 3g characterizing the number of vibrational quanta associated witheach Cartesian degree of freedom, and there are a set of annihilation, creation, and numberoperators aº , a+º ; and Nº for each axis. The di¤erent annihilation and creation operatorsobey characteristic commutation relations
[aº ; aº0 ] = 0 =£a+º ; a
+º0¤
(4.121)£aº ; a
+º0¤= ±º;º0 (4.122)
that, as we have seen, completely determine the associated integer spectrum of the numberoperators Nº = a+º aº :
Within this context, we now note that product operators of the form a+x ay havethe e¤ect
a+x ay fjnx; ny; nzig =pnx + 1
pny fjnx + 1; ny ¡ 1; nzig (4.123)
of transferring a quantum of vibrational excitation from one axis to another, i.e., ofinducing transitions in the states of the quanta. It is precisely operators of this type,that are capable of creating, destroying, and counting material particles in di¤erent singleparticle states that we wish to de…ne for a collection of identical particles.
In order to carry this plan out, however, we need to realize that these creation andannhilation operators have the e¤ect of taking the system out of the space of N -particles,and into a space containing N +1; or N ¡1 particles. Thus, we need to expand our spacein a way that allows us to put together, in the same space, states of the system containingdi¤erent particle numbers. The mathematical procedure for doing this is to combine thedi¤erent N-particle spaces together in what is referred to as a direct sum of vector spaceswhich is, in a way, similar to that associated with the direct product of vector spaces thatwe have already encountered.. The de…nition and details associated with this procedureare explored in the next section.
4.4.4 Fock Space as a Direct Sum of Vector Spaces
The idea of expressing a vector space as a sum of smaller vector spaces is actually implicitlycontained in some of the concepts that we have already encountered. Consider, e.g., an
148 Many Particle Systems
arbitrary observable A of a linear vector space S: By de…nition, the observable A possessesa complete orthonormal basis fja; ¿ig of eigenstates spanning the space. The eigenstatesassociated with a particular eigenvalue a form a subspace Sa of S; the vectors of whichare orthogonal to the vectors in any of the other eigenspaces of A: Also, any vector jÃiin S can be written as a linear combination of vectors taken from each of the orthogonalsubspaces Sa; i.e.,
jÃi =Xa
jÃai (4.124)
wherejÃai =
X¿
Ãa;¿ ja; ¿i = PajÃi: (4.125)
Under these circumstances, we say that the space S can be decomposed into a directsum of the eigenspaces associated with the observable A; and symbolically represent thisdecomposition in the form
S = Sa © Sa0 © Sa00 © ¢ ¢ ¢ (4.126)
The vector space S has a dimension equal to the sum of the dimensions of all theeigenspaces Sa; as can be seen by counting up the basis vectors needed to span eachorthogonal subspace.
This idea of decomposing larger spaces into direct sums of smaller spaces can bereversed. Thus, given two separate vector spaces S1 and S2 of dimension N1 and N2;respectively, de…ned on the same …eld of scalars, we produce a larger vector space S ofdimension N = N1 +N2 as the direct sum
S = S1 © S2: (4.127)
The space S then contains, by de…nition, all vectors in S1; all vectors in S2; and allpossible linear combinations of the vectors in S1 and S2 (the latter two spaces now beingrelegated to the role of orthogonal subspaces of S), with each vector in S1 orthogonal, byconstruction, to each vector in S2:
It is this procedure that we carry out with the di¤erent N -particle spaces associ-ated with a collection of identical particles. In particular, we de…ne the Fock Space ofa collection of identical bosons or fermions, respectively, as the space obtained by form-ing the direct sum of the corresponding spaces describing N = 0; 1; 2; ¢ ¢ ¢ particles. Inparticular, the Fock space of a set of identical bosons is written as the direct sum
SS = S0S © S1S © S2S © S3S © ¢ ¢ ¢ (4.128)
of the symmetric spaces S(N)S for each possible value of N: The space S0S of zero particlesis assumed to contain exactly one linearly-independent basis vector, referred to as thevacuum and denoted by j0i; and vectors from di¤erent N -particle spaces are assumed tobe automatically orthogonal to each other.
Similarly, the Fock space associated with a collection of identical fermions iswritten as the direct sum
SA = S0A © S1A © S2A © S3A © ¢ ¢ ¢ (4.129)
of the antisymmetric spaces S(N)S for each value of N; where, again, the space of zeroparticles contains one basis vector, the vacuum, denoted by j0i; and the di¤erent N -particle subspaces are assumed orthogonal. In what follows we explore separately thedi¤erent structure of the bosonic and fermionic Fock spaces.
Systems of Identical Particles 149
4.4.5 The Fock Space of Identical Bosons
As we have seen, the symmetric occupation number states
jn1; n2; : : :iXº
nº = N (4.130)
associated with a given set of single-particle states jÁºi forms an ONB for the space S(N)S
of N identical bosons. The collection of such states, therefore, with no restriction on thesum of the occupation numbers, forms a basis for the Fock space SS of a set of identicalbosons. In particular, the vacuum state is associated, in this (and any) occupation numberrepresentation with the vector
j0i = j0; 0; : : :i (4.131)
in which nº = 0 for all single particle states Áº . The single-particle states jÁºi themselvescan be expressed in this representation in the form
jÁ1i = j1; 0; 0; : : :ijÁ2i = j0; 1; 0; : : :i
... (4.132)
and so on. These basis states of Fock space are all simultaneous eigenvectors of thenumber operators Nº associated with this set of single particle states, and are uniquelylabeled by the associated set of occupation number fnºg : Thus, the operators fNºg forma CSCO for Fock space, just as they do for each of the symmetric N-particle subspacesS(N)S . Thus, we can write a completeness relation for this representation of the form
1Xn1=0
1Xn2=0
¢ ¢ ¢ jn1; n2; : : :ihn1; n2; : : : j = 1: (4.133)
We now wish to introduce operators that change the occupation numbers in a way similarto that associated with the harmonic oscillator. Because we are interested at present indescribing bosons we want the spectrum of each number operator Nº to be exactly thesame as for the harmonic oscillator, i.e., nº 2 f0; 1; 2; ¢ ¢ ¢ g : Thus, we can actually modelthe bosonic creation and annihilation operators directly on those of the oscillator system,i.e., we introduce for each single particle state jÁºi an annihilation and creation operator,whose action is de…ned on the occupation number states of this representation such that
a+º jn1; : : : ; nº ; : : :i =pnº + 1jn1; : : : ; nº + 1; : : :i (4.134)
andaº jn1; : : : ; nº ; : : :i = pnº jn1; : : : ; nº ¡ 1; : : :i: (4.135)
It follows from this de…nition, as special cases, that the single-particle state jÁºi (whichlies in the subspace containing just one boson) can be written in a form
jÁºi = a+º j0i = a+º j0; 0; : : :i (4.136)
in which it is created “from nothing” by the operator a+º : It also follows that any annihi-lation operator acting on the vacuum
aº j0i = 0 (4.137)
destroys it, i.e., maps it onto the null vector (not the vacuum!). Thus, the operatora+º creates a boson in the state Áº and the operator aº removes a boson from that state.
150 Many Particle Systems
Expressing the same idea in a somewhat more pedestrian fashion, the operators aº and a+ºsimply connect states in adjacent subspaces S(N)S of the bosonic Fock space SS. Consistentwith the de…nitions above, annihilation and creation operators associated with di¤erentsingle particle states commute with one another, allowing us to write the commutationrelations for the complete set of such operators in the form
[aº ; aº0 ] = 0 =£a+º ; a
+º0¤
(4.138)£aº ; a
+º0¤= ±º;º0 ; (4.139)
which are usually referred to as “boson commutation relations”. It also follows from thede…nition given above that
a+º aº jn1; : : : ; nº ; : : :i = nº jn1; : : : ; nº ; : : :i (4.140)
which allows us to identify this product of operators with the associated number operator,i.e., Nº = a+º aº ; which obey the commutation relations
[Nº ; Nº0 ] = 0 (4.141)
[Nº ; aº0 ] = ¡aº±º;º0£Nº ; a
+º0¤= a+º ±º;º0 (4.142)
Finally, just as it is possible to create the single particle state jÁºi from thevacuum, so it is possible to express any of the occupation number states of this represen-tation in a form in which they are created out of the vacuum by an appropriate productof creation operators, i.e., it is readily veri…ed that
jn1; : : : ; nº ; : : :i = (a+1 )n1(a+2 )
n2(a+3 )n3 ¢ ¢ ¢p
n1!pn2!pn3! ¢ ¢ ¢
j0i; (4.143)
the normalization factors taking the same form, for each single particle state, as in thesimple harmonic oscillator.
4.4.6 The Fock Space of Identical Fermions
The construction of the Fock space of a set of identical fermions proceeds in a similarfashion, but some interesting di¤erences arise as a result of the antisymmetric structureof the states associated with this space. For the fermionic space, we know that theantisymmetric occupation number states
jn1; n2; : : :iXº
nº = N (4.144)
generated from a given set of single-particle states jÁºi forms an ONB for the space S(N)S ofN identical fermions, provided that, as required by the exclusion principle, all occupationnumbers only take the values nº = 0 or nº = 1. Lifting the restriction on the sum of theoccupation numbers, we obtain a basis for the Fock space SA of this set of fermions. Asin the bosonic case, the vacuum state is associated with the vector
j0i = j0; 0; : : :i (4.145)
in which all nº = 0, and the single-particle states jÁºi can be written asjÁ1i = j1; 0; 0; : : :ijÁ2i = j0; 1; 0; : : :i
... (4.146)
Systems of Identical Particles 151
and so on. These basis states jn1; n2; : : :i of the fermionic Fock space are also simultaneouseigenvectors of the operators Nº ; and are uniquely labeled by their occupation numbersfnºg : Thus, the operators fNºg form a CSCO for the fermionic Fock space as well. Thecompleteness relation di¤ers from that of the bosonic space only by the limits on thesummations involved, i.e.,
1Xn1=0
1Xn2=0
¢ ¢ ¢ jn1; n2; : : :ihn1; n2; : : : j = 1: (4.147)
As in the bosonic case, we now wish to introduce operators that change the occupationnumbers. Clearly, however, we cannot model the fermion annihilation and creation oper-ators directly on those of the harmonic oscillator, since to do so would result in a fermionnumber spectrum inappropriately identical to that of the bosons. In the fermion case werequire that the number operator have eigenvalues restricted to the set f0; 1g :
It is a remarkable fact that a very slight modi…cation to the commutation relationsassociated with the annihilation and creation operators of the harmonic oscillator yielda set of operators that have precisely the properties that we need. To partially motivatethis modi…cation we note that the commutator [A;B] of two operators simply gives, if itis known, a rule or prescription for reversing the order of a product of these two operatorswhenever it is convenient to do so. Thus if we know the operator [A;B] then we canreplace the operator AB wherever it appears with the operator BA + [A;B] : Any rulethat allows us to perform a similar reversal would serve the same purpose. For example,it sometimes occurs that the anticommutator fA;Bg of two operators, de…ned as thesum
fA;Bg = AB +BA (4.148)
rather than the di¤erence of the operator product taken in each order, is actually a simpleroperator than the commutator. Under these circumstances, we can use the anticommu-tator to replace AB whenever it occurs with the operator ¡BA+ fA;Bg :
With this in mind, we now consider an operator a and its adjoint a+ which obeyanticommutation relations that have the same structure as the commutation relations ofthose associated with the harmonic oscillator, i.e., suppose that
fa; ag = 0 = ©a+; a+ª (4.149)©a; a+
ª= 1: (4.150)
Let us also de…ne, as in the oscillator case, the positive operator N = a+a and let usassume that N has at least one nonzero eigenvector jºi; square normalized to unity, witheigenvalue º; i.e., N jºi = ºjºi: The following statements are then easily shown:
1. a2 = 0 = (a+)2
2. aa+ = 1¡N
3. Spectrum(N) = f0; 1g
4. If º = 0; then aj0i = 0 and a+j0i = j1i; i.e., a+j0i is a square normalized eigenvectorof N with eigenvalue 1:
5. If º = 1; then a+j1i = 0 and aj1i = j0i; i.e., aj1i is a square normalized eigenvectorof N with eigenvalue 0:
152 Many Particle Systems
The …rst item follows directly upon expanding the anticommutation relationsfa; ag = 0 = fa+; a+g. The second item follows from the last anticommutation relationfa; a+g = aa+ + a+a = 1; which implies that aa+ = 1 ¡ a+a = 1 ¡ N: To show thatthe spectrum can only contain the values 0 an 1; we multiply the relation aa++ a+a = 1by N = a+a and use the fact that aa = 0 to obtain N2 = N; which implies that N is aprojection operator and so can only have the eigenvalues speci…ed . To show that both ofthese actually occur, we prove the last two items, which together show that if one of theeigenvalues occurs in the spectrum of N then so does the other. These statements followfrom the following observations
jjaj0ijj2 = h0ja+aj0i = h0jN j0i = 0 (4.151)
jja+j0ijj2 = h0jaa+j0i = h0j (1¡N) j0i = h0j0i = 1 (4.152)
Na+j0i = a+aa+j0i = a+ (1¡N) j0i= a+j0i (4.153)
which prove the fourth item, and
jja+j1ijj2 = h1jaa+j1i = h1j (1¡N) j1i = 0 (4.154)
jjaj1ijj2 = h1ja+aj1i = h1jN j1i = h1j1i = 1 (4.155)
Naj1i = a+aaj1i = 0 (4.156)
which proves the …nal item.It should be clear that operators of this type have precisely the properties we
require for creating and destroying fermions, since they only allow for occupation numbersof 0 and 1. We thus de…ne, for the fermion Fock space, a complete set of annihilation,creation, and number operators aº ; a+º ; and Nº = a
+º aº that remove, create, and count
fermions in the single particle states jÁºi: To de…ne these operators completely, we needto specify the commutation properties obeyed by annihilation and creation operatorsassociated with di¤erent single particle states. It turns out the antisymmetry of the statesin this space under particle exchange require that operators associated with di¤erent statesanticommute, so that, collectively, the operators associated with this occupation numberrepresentation obey the following fermion commutation relations
faº ; aº0g = 0 =©a+º ; a
+º0ª
©aº ; a
+º0ª= ±º;º0 ; (4.157)
which are just like those for bosons, except for replacement of the commutator bracketwith the anticommutator bracket. It is importnat to emphasize that, according to thesede…ntions, operators associated with di¤erent single-particle states do not commute, theyanticommute, which means for example that
a+º a+º0 = ¡a+º0a+º : (4.158)
Thus, the order in which particles are created in or removed from various states makes adi¤erence. This reversal of sign is reminiscent of, and stems from the same source, as thesign change that occurs when di¤erent direct product states in S(N) are projected intothe anti-symmetric subspace, i..e., it arises from the antisymmetry exhibited by the statesunder particle exchange.
Systems of Identical Particles 153
The action of these fermion annihilation and creation operators on the vacuum isessentially the same as for the boson operators, i.e.,
a+º j0i = jÁºi aº j0i = 0; (4.159)
and in a similar fashion we can represent an arbitrary occupation number state of thisrepresentation in terms of the vacuum state through the expression
jn1; : : : ; nº ; : : :i = (a+1 )n1(a+2 )n2(a+3 )n3 ¢ ¢ ¢ j0i: (4.160)
Note that normalization factors are not necessary in this expression, since all the occupa-tion numbers are equal to either 0 or 1: Also, the order of the creation operators in theexpression is important, with higher occupied states (states with large values of º) …lled…rst, since the creation operators for such states are closer to the vacuum state being actedupon than those with lower indices. This way of representing the occupation number statejn1; : : : ; nº ; : : :i as an ordered array of creation operators acting on the vacuum is oftenreferred to as standard form.
When an annihilation or creation operator acts upon an arbitrary occupationnumber state the result depends upon the whether the corresponding single particle statesis already occupied, as well as on the number and kind of single particle states alreadyoccupied. Speci…cally, it follows from the anticommutation relations above that
aº jn1; : : : ; nº ; : : :i =8<: 0 if nº = 0
(¡1)m jn1; : : : ; nº ¡ 1; : : :i if nº = 1(4.161)
and
a+º jn1; : : : ; nº ; : : :i =8<: (¡1)m jn1; : : : ; nº + 1; : : :i if nº = 0
0 if nº = 1(4.162)
wherem =
Xº0<º
nº0 (4.163)
are the number of states with º0 < º already occupied. The phase factors (¡1)m areeasily proven using the anticommutation relations. For example, the action of a+º on thestate jn1; : : : ; nº ; : : :i can be determined by expressing the latter in standard form, andthen anticommuting the creation operator through those with indices less than º until itis sitting in its standard position. Each anticommutation past another creation operatorgenerates a factor of ¡1; since a+º a+º0 = ¡a+º0a+º : The product of all these factors gives ¡1raised to the power m =
Pº0<º nº0 :
It is worth noting that, although the fermion creation and annihilation operatorsanticommute, the corresponding number operators Nº = a+º aº ; actually commute withone another. This follows from the basic de…nition of these operators in terms of theiraction on the occupation number states, but is also readily obtained using the anticom-mutation relations. To see this, consider
NºNº0 = a+º aºa
+º0aº0 with º 6= º0; (4.164)
and note that each time a+º0 is moved to the left one position it incurs a minus sign. Whenit moves two positions, all the way to the left, we have a product of two minus signs, soNºNº0 = a
+º0a
+º aºaº0 : But now we do the same thing with aº , moving it two positions to
the left and so …nd thatNºNº0 = a
+º0aº0a
+º aº = Nº0Nº : (4.165)
154 Many Particle Systems
In a similar fashion, e.g., we deduce, using the anticommutator aº0a+º + a+º aº0 = ±º;º0 ;
that
Nºaº0 = a+º aºaº0 = ¡a+º aº0aº= a+º0a
+º aº0 ¡ aº±º;º0 = aº0(Nº ¡ ±º;º0) (4.166)
which shows that[Nº ; aº0 ] = ¡aº0±º;º0 (4.167)
which is the same commutation relation as for bosons. In a similar fashion it is readilyestablished that aºa+º0 = ±ºº0 ¡ a+º0aº
Nºa+º0 = a+º aºa
+º0 = a
+º (±ºº0 ¡ a+º0aº)
= a+º0±ºº0 + a+º0a
+º aº = a
+º0±ºº0 + a
+º0Nº (4.168)
which shows that £Nº ; a
+º0¤= +a+º0±º;º0 (4.169)
also as for bosons.
4.4.7 Observables of a System of Identical Particles Revisited
Having compiled an appropriate set of operators capable of describing transitions betweendi¤erent occupation number states we now reconsider the form that general observablesof a system of identical particles take when expressed as operators of Fock space. Recallthat the problem that led to our introduction of Fock space was basically that di¤erentparts of the same operator (e.g. the Hamiltonian) are expressible as simple functions ofnumber operators in di¤erent occupation number representations. The problem is similarto that encountered in simpler quantum mechanical problems where the question oftenarises as to whether to work in the position representation, the momentum representation,or some other representation altogether. We are thus led to consider how the occupationnumber representations associated with di¤erent sets of single particle states are relatedto one another.
Recall that any orthonormal basis of single-particle states generates its own oc-cupation number representation. Thus, e.g. an ONB of states fjÁºi j º = 1; 2; ¢ ¢ ¢ ggenerates a representation of states jn1; : : : ; nº ; : : :i that are expressible in terms of a setof creation, annihilation, and number operators a+º ; aº ; and Nº = a+º aº ; while a dif-ferent ONB of states fj¹i j ¹ = 1; 2; ¢ ¢ ¢ g generates a di¤erent representation of statesj~n1; : : : ; ~n¹; : : :i; say, expressible in terms of a di¤erent set of creation, annihilation, andnumber operators b+¹ ; b¹; and Nº = b
+¹ b¹:We know, on the other hand, that the two sets
of single-particle states are related to one another through a unitary transformation Usuch that, e.g.,
jÁºi = U jºi º = 1; 2; ¢ ¢ ¢ (4.170)
with matrix elementsU¹º = h¹jU jºi = h¹jÁºi (4.171)
that are the inner products of one basis set in terms of the other. This allows us to write,e.g., that
j¹i =Xº
jÁºihÁº j¹i =Xº
U¤¹º jÁºi (4.172)
where U¤¹º = h¹jÁºi¤ = hÁº j¹i: But we also know that the single-particle states jÁºican be expressed in terms of the vacuum state through the relation
jÁºi = a+º j0i: (4.173)
Systems of Identical Particles 155
Using this in the expression for j¹i; we …nd that
j¹i =Xº
U¤¹º jÁºi =Xº
U¤¹ºa+º j0i
= b+¹ j0i (4.174)
where we have identi…ed the operator
b+¹ =Xº
U¤¹ºa+º (4.175)
that creates j¹i out of the vacuum. The adjoint of this relation gives the correspondingannihilation operator
b¹ =Xº
U¹ºaº : (4.176)
Thus the annihilation/creation operators of one occupation number representation arelinear combinations of the annihilation/creation operators associated with any other oc-cupation number representation, with the coe¢cients being the matrix elements of theunitary transformation connecting the two sets of single particle states involved. It isstraightforward to show that this unitary transformation of annihilation and creation op-erators preserves the boson or fermion commuation relation that must be obeyed for eachtype of particle. To treat both types of particles simultaneously, we introduce the nota-tion [A;B]§ = AB §BA; where the minus sign stands for the commutator (and appliesto boson operators) and the plus sign stands for the anticommutator (which applies tofermion operators). Thus, if the operators aº and a+º obey the relations
[aº ; aº0 ]§ = 0 =£a+º ; a
+º0¤
£aº ; a
+º0¤§ = ±º;º0 (4.177)
Then, using the transformation law we can calculate the corresponding relation for theoperators b¹ and b+¹ : Thus, e.g.,
[b¹; b¹0 ]§ = b¹b¹0 § b¹0b¹=
Xº;º0
U¹ºU¹0º0(aºaº0 § aº0aº)
=Xº;º0
U¹ºU¹0º0 [aº ; aº0 ]§ = 0 (4.178)
where in the last line we have used the appropriate relations for each type of particle.The adjoint of this relation shows that
hb+¹ ; b
+¹0
i§= 0; as well. In a similar fashion we see
that hb¹; b
+¹0
i§
=Xº;º0
U¹ºU¤¹0º0
£aº ; a
+º0¤§ =
Xº;º0
U¹ºU¤¹0º0±º;º0 =
Xº
U¹ºU¤¹0º
=Xº
h¹jÁºihÁº j¹0i = h¹j¹0i = ±¹;¹0 (4.179)
where we have used the completeness of the states jÁºi and the orthonormality of thestates j¹i: Thus, the b’s and b+’s obey the same kind of commutation/anticommutationrelations as the a’s and a+’s.
156 Many Particle Systems
We are now in a position to see what di¤erent one-body and two-body operatorslook like in various representations. Suppose, e.g., that H1 is a one-body operator that isrepresented in the occupation number states generated by the single-particle states j¹iin the form
H1 =X¹
"¹N¹ =X¹
"¹b+¹ b¹: (4.180)
This implies, e.g., that in the space of single-particle, the states j¹i are the associatedeigenstates of H1; i.e., H1j¹i = "¹j¹i: To …nd the form that this takes in any otheroccupation number representation we simply have to express the annihilation and creationoperators in (4.180) as the appropriate linear combinations of the new annihilation andcreation operators, i.e.,
H1 =Xº;º0¹
"¹U¤¹ºU¹0º0a
+º aº0
=Xº;º0
a+º Hºº0aº0 (4.181)
where
Hºº0 =X¹
U¤¹º"¹U¹0º0 =X¹
hÁº j¹i"¹h¹jÁº0i
= hÁº jH(1)1 jÁº0i (4.182)
in which we have identi…ed the expansion
H(1)1 =
X¹
j¹i"¹h¹j (4.183)
of the operator H1; as it is de…ned in the one-particle subspace. Although we haveexpressed this in a notation suggestive of Hamiltonians and energy eigenstates, the sameconsiderations apply to any one-body operator. Thus, a general one body operator can berepresented in Fock space in an arbitrary occupation number representation in the form
B1 =Xº;º0
a+º Bºº0aº0 Bºº0 = hÁº jB(1)1 jÁº0i: (4.184)
In the special case when B is diagonal in the speci…ed single particle representation, thedouble sum collapses into a single sum, and the resulting operator is reduced to a simplefunction of the number operators of that representation. Thus, a one-body operator Binduces single particle transitions, taking a particle out of state Áº0 and putting it intostate Áº with amplitude Bºº0 :
As we noted earlier, two body operators are often associated with interactionsbetween particles. Often, a representation can be found in which such an operator can beexpressed in the following form
H2 =1
2
X¹;¹0
¹6=¹0
N¹N¹0V¹¹0 +1
2
X¹
N¹(N¹ ¡ 1)V¹¹ (4.185)
where in the …rst term V¹¹0 is the interaction energy between a particle in the state j¹iand another particle in a di¤erent state j¹0i: The second term includes the interactionsbetween particles in the same states, and takes this form because a particle in a given
Systems of Identical Particles 157
state does not interact with itself (Note that 12N¹(N¹¡ 1) is the number of distinct pairsof N¹ particles). Both terms can be combined by inserting an appropriate Kroneckerdelta, i.e.,
H2 =1
2
X¹;¹0
N¹(N¹0 ¡ ±¹¹0)V¹¹0 : (4.186)
This can be simpli…ed further. Using the commutation laws [N¹0 ; b¹] = ¡b¹±¹¹0 obeyedby both fermion and boson operators (see the discussion in the last section) it follows that
N¹0b¹ = b¹(N¹0 ¡ ±¹¹0): (4.187)
Multiplying this on the left by b+¹ ; we deduce that
b+¹N¹0b¹ = b+¹ b
+¹0b¹0b¹ = N¹(N¹0 ¡ ±¹¹0); (4.188)
which allows us to write
H2 =1
2
X¹;¹0
b+¹ b+¹0V¹¹0b¹0b¹: (4.189)
Thus, in this form the two-body interaction is a sum of products involving two annihila-tion and two creation operators, but it only involves two summation indices, since eachannihilation operator is paired o¤ with a creation operator of each type.
To see what this looks like in any other occupation number representation we justhave to transform each of the annihilation and creation operators in the sum. Thus we…nd that in a representation associated with a set of states jÁºi = U jºi;
H2 =1
2
X¹;¹0
b+¹ b+¹0V¹¹0b¹0b¹
=1
2
Xq;r;s;t
X¹;¹0
U¤¹qU¤¹0rU¹0sU¹tV¹¹0a
+q a
+r0asat: (4.190)
where we have used the roman indices q; r; s; and t to avoid the proliferation of multiply-primed º’s. To simplify this we now re-express the matrix elements of the unitary trans-formation in terms of the inner products between basis vectorsX
¹;¹0U¤¹qU
¤¹0rU¹0sU¹tV¹¹0 =
X¹;¹0hÁqj¹ihÁrj¹0iV¹¹0h¹0 jÁsih¹jÁti (4.191)
and notice that each pair of inner products on the right and left of V¹¹0 can be expressed asa single inner product between direct product states in the space S(2) of just two particles,i.e.,X
¹;¹0hÁqj¹ihÁrj¹0iV¹¹0h¹0 jÁsih¹jÁti =
X¹;¹0hÁq; Árj¹; ¹0iV¹¹0h¹; ¹0 jÁt; Ási
= hÁq; ÁrjH(2)2 jÁt; Ási (4.192)
where the order of the terms has been chosen to reproduce the original set of four innerproducts, and where we have identi…ed
H(2)2 =
X¹;¹0
j¹; ¹0iV¹¹0h¹; ¹0 j (4.193)
158 Many Particle Systems
as the form that this operator takes in the space S(2) of just two particles. Working ourway back up, we …nd that in an arbitrary occupation number representation, a generaltwo-body interaction can be written in the form
H2 =1
2
Xq;r;s;t
a+q a+r Vqrtsasat (4.194)
whereVqrts = hÁq; ÁrjH(2)
2 jÁt; Ási (4.195)
is the matrix element of the operator taken between states of just two (distinguishable)particles. Thus, to construct such an operator for an arbitrary occupation number rep-resentation we simply need to be able to takes its matrix elements with respect to thecorresponding set of two-particle direct-product states.
As an example, we note that the Coulomb interaction
V (~r1; ~r2) =e
j~r1 ¡ ~r2j (4.196)
between particles can be written in the form
V =1
2
Xq;r;s;t
a+q a+r0Vqrtsasat (4.197)
where the matrix elements are evaluated, e.g., in the two-particle position representationas
Vqrts = hÁq; ÁrjV jÁt; Ási=
Zd3r
Zd3r0 Á¤q(~r)Á
¤r(~r
0)e
j~r ¡ ~r 0jÁt(~r)Ás(~r0) (4.198)
in terms of the wave functions associated with this set of single particle states.
4.4.8 Field Operators and Second Quantization
The use of creation and annihilation operators of the type we have just considered is oftenreferred to as the method of second quantization . The “…rst quantization” impliedby this phrase is that developed, e.g., by Schrödinger, in which the dynamical variablesxi and pi of classical mechanics are now viewed as operators, and the state of the systemis characterized by wave functions Ã(xi; t) or Ã(pi; t) of one or another of this set ofvariables. On the other hand, there are other systems studied by classical mechanics thatcannot be described as particles, e.g., waves traveling on a string or through an elasticmedium, or Maxwell’s electric and magnetic …eld equations, where the classical dynamicalvariables are just the …eld amplitudes Ã(x; t) at each point, which like the position andmomentum of classical particles always have a well-de…ned value, and where x is nowsimply viewed as a continuous index labeling the di¤erent dynamical variables Ã(x) thatare collectively needed to fully describe the con…guration of the system. In a certainsense, Schrödinger’s wave function Ã(x; t) for a single particle can also be viewed “as asort of classical …eld” and Schrödinger’s equation of motion can be viewed as simply the“classical” wave equation obeyed by this …eld. One can then ask what happens when thisclassical system is quantized, with the corresponding …eld amplitudes being associatedwith operators. It is actually possible to follow this path from classical …elds to quantumones through a detailed study of the objects of classical …eld theory, including Lagrangiandensities, conjugate …elds, and so on.
Systems of Identical Particles 159
As it turns out, however, the end result of such a process is actually implicitlycontained in the mathematical developments that we have already encountered. Thekey to seeing this comes from the realization that the procedure for transforming be-tween di¤erent occupation number representations applies, in principle, to any two setsof single-particle states. Until now we have focused on transformations between discreteONB’s, e.g., fjÁºig and
©j¹iª ; but it is possible to consider transformations that includecontinuously indexed basis sets as well.
A case of obvious interest would be the basis states fj~rig of the position repre-sentation. The transformation law between these states and those of some other single-particle representation, e.g., fjÁºig takes the form
jÁºi =Zd3r j~rih~rjÁºi =
Zd3r Áº(~r) j~ri =
Zd3r Uº(~r) j~ri (4.199)
andj~ri =
Xº
jÁºihÁº j~ri =Xº
Á¤º(~r)jÁºi =Xº
U¤º (~r)jÁºi (4.200)
where Uº(~r) = h~rjÁºi and U¤º (~r) = hÁº j~ri are simply the wave functions (and conjugates)for the single particle states jÁºi in the position representation. Expressing, the single-particle state jÁºi in terms of the vacuum state, and substituting into the expansion forthe state j~ri; we …nd that
j~ri =Xº
Á¤º(~r)a+º j0i: (4.201)
This allows us to identify the operator that creates out of the vaccum a particle at thepoint ~r; (i.e., in the single-particle state j~ri) as a linear combination of creation operatorsassociated with the states jÁºi. We will denote this new creation operator with the symbolÃ+(~r); i.e.,
Ã+(~r) =
Xº
Á¤º(~r)a+º : (4.202)
The adjoint of the operator Ã+(~r) gives the corresponding annihilation operator
Ã(~r) =Xº
Áº(~r)aº (4.203)
These two families of operators are referred to as …eld operators, since they de…ne anoperator-valued …eld of the real space position parameter ~r. Thus Ã
+(~r) creates a par-
ticle at ~r; and Ã(~r) destroys or removes a particle from that point. The fact that thebasis states j~ri of this single-particle representation are not square-normalizable leads tosome slight but fairly predictable di¤erences between the …eld operators and the anni-hilation and creation operators associated with discrete representations. For example,the commutation/anticommutation relations obeyed by the …eld operators now take aform more appropriate to the continuous index associated with this set of operators. Thetransformation law is derived as in the discrete case, and we …nd thath
Ã(~r); Ã(~r 0)i§=Xº;º0
Áº(~r)Áº0(~r0) [aº ; aº0 ]§ = 0 (4.204)
hÃ+(~r); Ã
+(~r 0)
i§=Xº;º0
Á¤º(~r)Á¤º0(~r
0)£a+º ; a
+º0¤§ = 0 (4.205)
160 Many Particle Systems
and hÃ(~r); Ã
+(~r 0)
i§
=Xº;º0
Áº(~r)Á¤º0(~r
0)£a+º ; a
+º0¤§
=Xº;º0
Áº(~r)Á¤º0(~r
0)±ºº0 =Xº;º
Áº(~r)Á¤º(~r
0)
=Xº;º
h~rjÁºihÁº j~r 0i = h~rj~r 0i
= ±(~r ¡ ~r 0) (4.206)
Also, as a consequence of the normalization, the product of Ã+(~r) and Ã(~r) does not give
a number operator, but a number density operator, i.e.,
n(~r) = Ã+(~r)Ã(~r) (4.207)
counts the number of particles per unit volume at the point ~r. That the product has thecorrect units to describe a number density follows directly from the commutation relationsjust derived. From this we can de…ne number operators N that count the number ofparticles in any region of space ; as the integral
N =
Z~r2
d3r Ã+(~r)Ã(~r) (4.208)
with the total number operator N obtained by extending the integral to all space.Finally, we can express various one-body and two body operators using this rep-
resentation, by sightly extending our results obtained with discrete representations. Thus,e.g., a general one-body operator can be expressed in terms of the …eld operators throughthe expression
H1 =
Zd3r
Zd3r0 Ã
+(~r)H1(~r; ~r
0)Ã(~r) H1(~r;~r0) = h~rjH(1)
1 j~r 0i: (4.209)
For a collection of noninteracting identical particles moving in a common potential, e.g.,
H(~r;~r 0) = h~rjP2
2mj~r 0i+ h~rjV j~r 0i = ¡ ~
2
2mr2±(~r ¡ ~r 0) + V (~r)±(~r ¡ ~r 0) (4.210)
which reduces the previous expression to the familiar looking form
H =
Zd3r
·Ã+(~r)
µ¡ ~
2
2m
¶r2Ã(~r) + Ã+(~r)V (~r)Ã(~r)
¸(4.211)
which, it is to be emphasized is an operator in Fock space, although it looks just like asimple expectation value. Similarly, a general two-body operator can be expressed in thesomewhat more cumbersome form
H2 =
Zd3r1
Zd3r2
Zd3r3
Zd3r4 Ã
+(~r1)Ã
+(~r2)h~r1; ~r2jH2j~r3; ~r4iÃ(~r4)Ã(~r3): (4.212)
The form that this takes for the Coulomb interaction is left as an exercise for the reader.
Chapter 5APPROXIMATION METHODS FOR STATIONARY STATES
As we have seen, the task of prediciting the evolution of an isolated quantum mechan-ical can be reduced to the solution of an appropriate eigenvalue equation involving theHamiltonian of the system. Unfortunately, only a small number of quantum mechanicalsystems are amenable to an exact solution. Moreover, even when an exact solution tothe eigenvalue problem is available, it is often useful to understand the behavior of thesystem in the presence of weak external fields that my be imposed in order to probe thestructure of its stationary states. In these situations an approximate method is requiredfor calculating the eigenstates of the Hamiltonian in the presence of a perturbation thatrenders an exact solution untenable. There are two general approaches commonly takento solve problems of this sort. The first, referred to as the variational method, is mostuseful in obtaining information about the ground state of the system, while the second,generally referred to as time-independent perturbation theory, is applicable to any set ofdiscrete levels and is not necessarily restricted to the solution of the energy eigenvalueproblem, but can be applied to any observable with a discrete spectrum.
5.1 The Variational Method
Let H be a time-independent observable (e.g., the Hamiltonian) for a physical systemhaving (for convenience) a discrete spectrum. The normalized eignestates |φni of Heach satisfy the eigenvalue equation
H|φni = En|φni (5.1)
where for convenience in what follows we assume that the eigenvalues and correspondingeigenstates have been ordered, so that
E0 ≤ E1 ≤ E2 · · · . (5.2)
Under these circumstances, if |ψi is an arbitrary normalized state of the system it isstraightforward to prove the following simple form of the variational theorem: themean value of H with respect to an arbitrary normalized state |ψi is necessarily greaterthan the actual ground state energy (i.e., lowest eigenvalue) of H, i.e.,
hHiψ = hψ|H|ψi ≥ E0. (5.3)
The proof follows almost trivially upon using the expansion
H =Xn
|φniEnhφn| (5.4)
of H in its own eigenstates to express the mean value of interest in the form
hHiψ =Xn
hψ|φniEnhφn|ψi =Xn
|ψn|2En, (5.5)
134 Approximation Methods for Stationary States
and then noting that each term in the sum is itself bounded, i.e., |ψn|2En ≥ |ψn|2E0, sothat X
n
|ψn|2En ≥Xn
|ψn|2E0 = E0 (5.6)
where we have used the assumed normalization hψ|ψi = Pn |ψn|2 = 1 of the otherwise
arbitrary state |ψi. Note that the equality holds only if |ψi is actually proportional to theground state of H.
Thus, the variational theorem proved above states that the ground state minimizesthe mean value of H taken with respect to the normalized states of the space. This hasinteresting implications. It means, for example, that one could simply choose randomvectors in the state space of the system and evaluate the mean value of H with respectto each. The smallest value obtained then gives an upper bound for the ground stateenergy of the system. By continuing this random, or “Monte Carlo”, search it is possible,in principle, to get systematically better (i.e., lower) estimates of the exact ground stateenergy.
It is also possible to prove a stronger statement that includes the simple boundgiven above as a special case: the mean value of H is actually stationary in the neigh-borhood of each of its eigenstates |φi. This fact, which is a more complete and precisestatement of the variational theorem, is compactly expressed in the language of the cal-culus of variations through the relation
δhHiφ = 0. (5.7)
To see what this means physically, let |φi be a normalizable state of the systemabout which we consider a family of kets
|φ(λ)i = |φi+ λ|ηi (5.8)
that differ by a small amount from the original state |φi, where |ηi is a fixed but arbitrarynormalizeable state and λ is a real parameter allowing us to parameterize the small butarbitrary variations
δ|φi = |φ(λ)i− |φi = λ|ηi (5.9)
of interest about the ket |φi = |φ(0)i.Let us now denote by
ε(λ) = hHiλ = hφ(λ)|H|φ(λ)ihφ(λ)|φ(λ)i (5.10)
the mean value of H with respect to the varied state |φ(λ)i, in which we have includedthe normalization in the denominator so that we do not have to worry about constrainingthe variation to normalized states. With these definitions, then, we wish to prove thefollowing: the state |φi is an eigenstate of H if and only if, for arbitrary |ηi,
∂ε
∂λ
¯λ=0
= 0. (5.11)
To prove the statement we first compute the derivative of ε(λ) using the chainrule, i.e., introducing the notation
|φ0i = ∂|φ(λ)i∂λ
= |ηi hφ0| = ∂hφ(λ)|∂λ
= hη| (5.12)
The Variational Method 135
we have (since H is independent of λ)
∂ε
∂λ
¯λ=0
=hφ0|H|φihφ|φi +
hφ|H|φ0ihφ|φi − hφ|H|φihφ|φi2
£hφ0|φi+ hφ|φ0i¤ . (5.13)
This can be multiplied through by hφ|φi and the identity |φ0i = |ηi used to obtain therelation
hφ|φi ∂ε∂λ
¯λ=0
= hη|H|φi+ hφ|H|ηi− hφ|H|φihφ|φi [hη|φi+ hφ|ηi] . (5.14)
Now denote by E the mean value of H with respect to the unvaried state, i.e., set
E = ε(0) =hφ|H|φihφ|φi (5.15)
and write E [hη|φi+ hφ|ηi] = hη|E|φi+ hφ|E|ηi to put the above expression in the form
hφ|φi ∂ε∂λ
¯λ=0
= hη|(H −E)|φi+ hφ|(H −E)|ηi. (5.16)
We now note that if |φi is an actual eigenstate of H its eigenvalue must be equal toE = ε(0), in which case the right hand side of the last expression vanishes (independentof the state |ηi). Since |φi is nonzero, we conclude that the derivative in (5.11) and (5.16)vanishes for arbitrary variations δ|φi = λ|ηi about any eigenstate |φi of H.
To prove the converse we note that, if the derivative of ε(λ) with respect to λdoes indeed vanish for arbitrary kets |ηi, then it must do so for any particular ket wechoose; for example, if we pick
|ηi = (H −E)|φi. (5.17)
then, (5.16) above reduces to the relation
0 = hη|(H −E)|φi+ hφ|(H −E)|ηi = hφ|(H −E)2|φi. (5.18)
Because, by assumption,H is Hermitian and E real we can now interpret this last equationas telling us that
hφ|(H −E)2|φi = ||(H −E)|φi||2 = 0, (5.19)
which means that the vector (H−E)|φimust vanish, and that |φi is therefore an eigenstateof H with eigenvalue E whenever the derivative (5.11) vanishes for arbitrary variationsδ|φi = λ|ηi, completing the proof.
In practice, use of this principle is referred to as the variational method, the basicsteps of which we enumerate below:
1. Choose an appropriate family |φ(α)i of normalized trial kets which depend pa-rameterically on a set of variables α = α1,α2, · · · ,αn , referred to as variationalparameters.
2. Calculate the mean value
hH(α)i = hφ(α)|H|φ(α)i (5.20)
as a function of the parameters α.
3. Minimize E(α) = hH(α)i with respect to the variational parameters by finding thevalues α0 for which
∂hHi∂αi
¯α=α0
= 0 i = 1, 2, · · · , n. (5.21)
136 Approximation Methods for Stationary States
The value E(α0) so obtained is the variational estimate of the ground state energywith respect to this family of trial kets, and the corresponding state |φ(α0)i provides thecorresponding variational approximation to the ground state.
It should be noted that if the family |φ(α)i of trial kets actually contains theground state (or any excited state), the variational principle shows that the techniquedescribed above will find it and the corresponding energy exactly. This can sometimesbe exploited. For example, if symmetry properties of the ground state are known (parity,angular momentum, etc.) it is often possible to choose a family of trial kets that areorthogonal to the exact ground state of the system. In this situation, the variationalmethod will then yield an upper bound to the energy of the lowest lying excited state ofthe system that is not orthogonal to the family of trial kets employed. We also note, inpassing, that our derivation of the variational principle shows that an approximation tothe ground state |φ0i that is correct to order ε (i.e., |φ(ε)i = |φ0i + ε|ηi) will yield anestimate of the ground state energy E0 which is correct to order ε2. This follows from thefact that in an expansion
E (ε) = E0 + ε∂E
∂ε
¯ε=0
+ε2
2!
∂2E
∂ε2
¯ε=0
+ · · · (5.22)
of the mean energy about that of the actual ground state, the linear term vanishes due tothe stationarity condition derived above. This explains the often observed phenomenonthat a rather poor approximation to the eigenstate can yield a relatively good estimateof the ground state energy.
A particulalry useful application of the variational method involves what is re-ferred to as the Rayleigh-Ritz method, which overcomes to some extent the usual difficultyof dealing with an infinite dimensional space. Suppose for example that we were to takeas a trial ket a state
|φi =Xi
φi|ii (5.23)
expanded in terms of some orthonormal set of vectors |ii, and take the expansioncoefficients φi (or their real and imaginary parts) as our variational parameters. If the set|ii is complete, then the family |φi of trial kets includes all physical vectors in the statespace, and the resulting variational procedure will just generate the exact eigenvectors ofH. Suppose, on the other hand, that the states |ii are not complete, but span someN−dimensional subspace SN . The variational procedure would then search through thisfinite dimensional subspace to find those states that are closest to being actual eigenstatesof the full system. The resulting vectors would then extremize the mean value
hHi = hφ|H|φi =Xi,j
φ∗iHijφj (5.24)
taken with respect to the states |φi in this subspace. In this last expession, Hij = hi|H|jidenotes the matrix elements of H with respect to the orthonormal states |ii spanningthe subspace SN . Suppose, however, we introduce a new Hermitian operator H(S) definedonly on the subspace SN and having the same matrix elements
H(S)ij = hi|H|ji = Hij (5.25)
as H within that subspace (but which vanishes outside of SN ); the N eigenvectors of thisrestricted operator H(S) will be those states |φi in SN which extremize the mean value
hH(S)i =Xi,j
φ∗iH(S)ij φj =
Xi,j
φ∗iHijφj , (5.26)
Perturbation Theory for Nondegenerate Levels 137
i.e., they will be precisely the variational eigenstates of the full Hamiltonian H that weare looking for. Thus, in this case, application of the variational method simply amountsto diagonalizing the matrix representing H restricted to some finite subspace of interest.Moreover, as implied by our previous comments on the variational method, the Rayleigh-Ritz method described above will exactly find any actual eigenvectors ofH that lie entirelywithin the chosen subspace SN .
5.2 Perturbation Theory for Nondegenerate Levels
We now turn to a more general and systematic method for determining the eigenvectorand eigenvalues for observables with a discrete spectrum. As in the last section we use thelanguage of energy eigenstates and Hamiltonia even though the method itself is perfectlyapplicable to other observables. For the purposes of stating the initial problem of interest,however, we consider the eigenvalue problem for a time-independent Hamiltonian
H = H(0) +H(1) = H(0) + λV (5.27)
having a discrete nondegenerate spectum. In writing the Hamiltonian in this form, theeigenvalue problem for the operator H(0), which will be referred to as the “unperturbedpart” of the Hamiltonian, is assumed to have been solved, and the perturbationH(1) = λVis presumed to be, in some sense, small compared toH(0). Our goal is to obtain expressionsfor the eigenstates |ni = |n(λ)i and eigenvalues εn = εn(λ) of H as an expansion in powersof the small, real parameter λ. These eigenstates of the full Hamiltonian H are to beexpressed as linear combinations and simple functions of the known eigenstates |n(0)i andeigenvalues ε(0)n of the unperturbed Hamiltonian H(0). Thus, the exact and unperturbedstates of the system are assumed to satisfy the equations
(H − εn)|ni = 0 hn|mi = δn,mXn
|nihn| = 1 (5.28)
(H(0) − ε(0)n )|n(0)i = 0 hn(0)|m(0)i = δn,mXn
|n(0)ihn(0)| = 1, (5.29)
the two relations on the right of the last two lines indicating that both sets of states forman ONB for the space of interest. We wish to identify, in particular, the unperturbedeigenstates |n(0)i as those to which the exact states |ni tend as λ → 0. This still leavesthe relative phase of the two sets of basis vector undetermined, as we could multiply thebasis vectors of one set by an arbitrary set of phases eiφn without affecting the validity ofthe equations above. For nonzero values of λ, therefore, we further fix the relative phasebetween these two sets of states by requiring that the inner product hn|n(0)i betweencorresponding elements of these two basis sets be real and positive.
Now, by assumption, there exist expansions of the full eigenstate |ni and thecorresponding eigenenergy εn of the form
|ni = |n(0)i+ λ|n(1)i+ λ2|n(2)i+ · · · (5.30)
εn = ε(0)n + λε(1)n + λ2ε(2)n + · · · (5.31)
We will refer to the terms λkε(k)n and λk|n(k)i as the kth order correction to the ntheigenenergy and eigenstate, respectively. The corresponding correction to the energy isalso generally referred to as the kth order energy shift, for obvious reasons. To deter-mine these corrections, we will simply require that the exact eigenstate |ni satisfy theappropriate eigenvalue equation
(H − εn)|ni = 0 (5.32)
138 Approximation Methods for Stationary States
to all orders in λ. Upon substitution of the expansions for |ni and εn into the eigenvalueequation we obtain
(H(0) + λV )∞Xk=0
λk|n(k)i =∞Xk=0
λkε(k)n
∞Xj=0
λj |n(j)i =∞X
k,j=0
λj+kε(k)n |n(j)i (5.33)
or∞Xk=0
λkH(0)|n(k)i+ V λk+1|n(k)i−∞Xj=0
λj+kε(k)n |n(k)i = 0. (5.34)
For this equation to hold for small but arbitrary values of λ, the coefficients of each powerof that parameter must vanish separately. The reason for this is essentially that thepolynomials fk(λ) = λk form a linearly independent set of functions on R, so any relationof the form
P∞k=0 ckλ
k = 0 can only be satisfied for all λ in R if ck = 0 for all k. Applyingthis requirement to the last equation generates an infinite heirarchy of coupled equations,one for each power of λ. The equation generated by setting the coefficient of λk equal tozero is referred to as the kth order equation. Collecting coefficients of the first few powersof λ we obtain after a little rearrangement the zeroth order equation
(H(0) − ε(0)n )|n(0)i = 0, (5.35)
the first order equation
(H(0) − ε(0)n )|n(1)i+ (V − ε(1)n )|n(0)i = 0, (5.36)
the second order equation
(H(0) − ε(0)n )|n(2)i+ (V − ε(1)n )|n(1)i− ε(2)n |n(0)i = 0, (5.37)
the third order equation
(H(0) − ε(0)n )|n(3)i+ (V − ε(1)n )|n(2)i− ε(2)n |n(1)i− ε(3)n |n(0)i = 0, (5.38)
and finally, after inspecting those which precede it, we deduce for k ≥ 2 the form of thegeneral kth order equation
(H(0) − ε(0)n )|n(k)i+ (V − ε(1)n )|n(k−1)i−kXj=2
ε(j)n |n(k−j)i = 0. (5.39)
As we will demonstrate, the structure of these equations allows for the general kth ordersolutions to be obtained from those solutions of lower order, allowing for the developmentof a systematic expansion of the eigenstates and eigenenergies in powers of λ. To beginthe demonstration we note that the zeroth order equation (5.35) is already satisfied, byassumption. From knowledge of the unperturbed states and eigenenergies, Eq. (5.36) canbe solved to give the first order correction ε
(1)n to the energy. This is most easily done by
simply multiplying (5.36) on the left by the unperturbed eigenbra hn(0)|, i.e.,hn(0)|(H(0) − ε(0)n )|n(1)i+ hn(0)|(V − ε(1)n )|n(0)i = 0. (5.40)
Since H(0) is Hermitian (and ε(0)n therefore real) it follows that hn(0)|(H(0) − ε
(0)n ) = 0,
and so the first order equation reduces to the relation
ε(1)n = hn(0)|V |n(0)i λε(1)n = hn(0)|H(1)|n(0)i. (5.41)
Perturbation Theory for Nondegenerate Levels 139
Thus, the first order correction to the energy eigenvalue for the nth level is simplythe mean value of the perturbing Hamiltonian taken with respect to the unperturbedeigenfunctions. Note that the first order correction to the energy comes from a meanvalue taken with respect to the zeroth order approximation to the state, consistent withthe remarks made earlier in the context of the variational method. As we will see, asimilar structure persists to all orders of perturbation theory, namely, an approximationof the state to kth order generates an approximation to the energy that is correct to orderk + 1.
Now that we have ε(1)n , we can put it back in to the first order equation (5.36) tofind an expansion for the first order correction |n(1)i to the eigenstate. Since we want toexpress this correction as an expansion
|n(1)i =Xm
|m(0)ihm(0)|n(1)i (5.42)
in unperturbed eigenstates |m(0)i of H(0), we obviously need to evaluate the expansioncoeffiicients hm(0)|n(1)i. Thus, we now take inner products of the first order equation(5.36) with the other members of this complete set of states, i..e., for the states withm 6= n. Multiplying (5.36) on the left by hm(0)| we obtain
hm(0)|(H(0) − ε(0)n )|n(1)i+ hm(0)|(V − ε(1)n )|n(0)i = 0 (5.43)
and observe that for the first term on the left of this expression
hm(0)|(H(0) − ε(0)n )|n(1)i = (ε(0)m − ε(0)n )hm(0)|n(1)i, (5.44)
while orthogonality of the unperturbed states implies that, for the second term,
hm(0)|ε(1)n |n(0)i = ε(1)n hm(0)|n(0)i = 0 for m 6= n. (5.45)
Thus, we obtain after a little rearrangement the following result
hm(0)|n(1)i = −hm(0)|V |n(0)iε(0)m − ε
(0)n
m 6= n. (5.46)
for the expansion coefficients of interest. This procedure for obtaining the expansioncoefficients for the state |n(1)i does not work for the term with m = n, since it just leads,again, to the expression ε
(1)n = hn(0)|V |n(0)i for the first order energy correction. As it
turns out, however, the one remaining expansion coefficient hn(0)|n(1)i can be evaluatedfrom the normalization condition hn|ni = 1 and our already chosen phase convention.The normalization condition implies the expansion
1 =hhn(0)|+ λhn(1)|+ λ2hn(2)|+ · · ·
i[|n(0)i+ λ|n(1)i+ λ2|n(2)i+ · · · ]
= hn(0)|n(0)i+ λhhn(1)|n(0)i+ hn(0)|n(1)i
i+O(λ2). (5.47)
Since, by assumption, hn(0)|n(0)i = 1, all of the remaining terms on the right-hand sideof this expansion must vanish, term-by-term. Thus, to first order normalization of theeigenstates requires that
0 = hn(1)|n(0)i+ hn(0)|n(1)i = 2Re³hn(0)|n(1)i
´. (5.48)
On the other hand, we have chosen our phase convention so that the inner product
hn|n(0)i =hhn(0)|+ λhn(1)|+ λ2hn(2)|+ · · ·
i|n(0)i
= hn(0)|n(0)i+ λhn(1)|n(0)i+ λ2hn(2)|n(0)i+ · · · (5.49)
140 Approximation Methods for Stationary States
is real and positive. Setting the imaginary part equal to zero gives the condition
∞Xk=1
λk Imhhn(k)|n(0)i
i= 0, (5.50)
which again requires, for arbitrary λ, that each inner product in the sum be separatelyreal. Combining this with (5.48) we deduce, therefore, that
hn(0)|n(1)i = 0. (5.51)
By our simple choice of phase, then, the first order correction |n(1)i is forced to be or-thogonal to the unperturbed eigenstate |n(0)i. Using this fact we then end up with thefollowing expansion
|n(1)i = −Xm6=n
|m(0)ihm(0)|V |n(0)iε(0)m − ε
(0)n
(5.52)
for the first order correction, and similar expansions
|ni = |n(0)i−Xm6=n
hm(0)|λV |n(0)iε(0)m − ε
(0)n
|m(0)i+O(λ2) (5.53)
|ni = |n(0)i−Xm6=n
hm(0)|H(1)|n(0)iε(0)m − ε
(0)n
|m(0)i+O(λ2) (5.54)
for the full eigenstate, correct to first order in the perturbation. This expression showsthat the pertubation H(1) “mixes” the eigenstates of H(0), by which is referred to thefact that the eigenstates of H are linear combinations of the unperturbed eigenstates.Note also that the presence of the “energy denominators” ε
(0)m − ε
(0)n appearing in the
expansion coefficients in this expression tend to mix together states close together inenergy more strongly than states that are energetically disparate. This makes it clearwhy we assumed from the outset that the unperturbed spectrum was non-degenerate,since the method we have developed clearly must fail when applied to perturbations thatconnect degenerate states. It is also clear that an implicit condition for the perturbationexpansion to converge, i.e., that the correction terms be sufficiently small is that
|hm(0)|H(1)|n(0)i| ¿¯ε(0)m − ε(0)n
¯(5.55)
for all states |m(0)i connected to the state |n(0)i by the perturbing Hamiltonian.When the first order correction to the energy vanishes, or higher accuracy is
required, it is necessary to go to higher order in the perturbation expansion. To obtainthe second order energy correction we proceed as follows: multiply the second orderequation (5.37) on the left by the unperturbed eigenbra hn(0)| to obtain
hn(0)|(H(0) − ε(0)n )|n(2)i+ hn(0)|(V − ε(1)n )|n(1)i− ε(2)n hn(0)|n(0)i = 0. (5.56)
Again using the fact that hn(0)| is an eigenbra of H0, along with the orthogonality relationhn(0)|n(1)i = 0 deduced above, we find that
ε(2)n = hn(0)|V |n(1)i. (5.57)
Inserting the expansion deduced above for |n(1)i, we then obtain the second order energyshift
λ2ε(2)n = −Xm6=n
hn(0)|λV |m(0)ihm(0)|λV |n(0)iε(0)m − ε
(0)n
= −Xm6=n
|λVmn|2ε(0)m − ε
(0)n
(5.58)
Perturbation Theory for Nondegenerate Levels 141
where the quantities λVmn = hm(0)|λV |n(0)i = H(1)mn are just the matrix elements of the
perturbation between the unperturbed eigenstates. Thus, the full eigenenergies, correctto second order, are given by the expression
εn = ε(0)n +H(1)nn −
Xm6=n
|H(1)mn|2
ε(0)m − ε
(0)n
+O(λ3). (5.59)
In problems involving weak perturbations it usually suffiices to determine corrections andenergy shifts to lowest non-vanishing order in the perturbation, and so it is unusal that oneneeds to go beyond second order for “simple” problems in perturbation theory. Exceptionsto this general observation arise quite often when dealing with many-body problems, wherediagramatic methods have been developed that take the ideas of perturbation theory toan extrememly high level, and where it is not uncommon to find examples where effectsof the perturbation are calculated to all orders.
It is worth pointing out, however, that the first order correction to the energycontains no information in it about any changes that occur in the eigenstates of the systemas a result of the perturbation. This information appears for the first time in the secondorder energy shift, as the derivation above makes clear. In many cases it is not possibleto perform the sum in (5.59) exactly, and so it is useful to develop simple means forestimating the magnitude of the second order energy shift. As it turns out, it is oftenstraightforward to develop upper and lower bounds for the magnitude of the change inenergy that occurs in any given eigenstate.
For example, a general upper bound for the second order shift can be obtained forany nondegenerate level by observing that
ε(2)n =Xm6=n
|H(1)mn|2
ε(0)n − ε
(0)m
≤Xm6=n
|H(1)mn|2¯
ε(0)n − ε
(0)m
¯ (5.60)
where in the right hand side we have a sum of positive definite terms that will alwaysbe larger in magnitude than a similar sum in which some of the corresponding terms arepositive and some negative, depending upon where the energy of each level lies relativeto the one of interest. Moreover, each term in the sum on the right can itself be bounded,since the energy denominators are bounded from below by that associated with the levelclosest in energy to the state |n(0)i. If we denote by ∆εn the energy spacing between leveln and the state closest in energy to it, then
¯ε(0)n − ε
(0)m
¯≥ ∆εn and so
ε(2)n ≤1
∆εn
Xm6=n
|H(1)mn|2 =
λ2
∆εn
Xm6=n
|Vmn|2
We can perform the infinite sum by “removing” the restriction on the summation index.We do this latter trick by adding and subtracting the quantity |Vnn|2 = |hV in|2 , wherehV in = Vnn = hn(0)|V |n(0)i is the mean value of the perturbation taken with respect tothe unperturbed eigenstate. (It is, essentially, just the first order energy correction ε
(1)n ).
Performing this operation allows us to write the upper bound above in the form
ε(2)n ≤λ2
∆εn
"Xm
|hm(0)|V |n(0)i|2 − |hV in|2#. (5.61)
where the sum is now unrestricted. But since the unperturbed states form a complete set
142 Approximation Methods for Stationary States
of states we can now rewrite the sum asXm
|hm(0)|V |n(0)i|2 =Xm
hn(0)|V |m(0)ihm(0)|V |n(0)i
= hn(0)|V |n(0)i = hV 2in (5.62)
which is just the mean value of the square of the perturbation taken with respect to theunperturbed eigenstate. Making this substition above and recognizing the root-mean-square statistical uncertainty
∆2V = hV 2i− hV i2 (5.63)
associated with the perturbing operator taken with respect to the unperturbed state ofinterest, we obtain our final result for the upper bound
ε(2)n ≤λ2∆2V
∆εn=∆2H(1)
∆εn(5.64)
in the second order energy shift. We note in passing that this gives another intuitivelyreasonable measure for determining the validity of the perturbation expansion, whichrequires for the smallness of ε(2)n that the uncertainty in H(1) be small relative to theenergy level spacing associated with the unperturbed states, i.e., that ∆H(1)/∆εn << 1.
For the ground state energy (or more generally the extremal eigenvalue) it is alsopossible to determine a lower bound on the magnitude of the second order energy shift.For the ground state, such a bound follows from the fact that in this case the second orderenergy shift
ε(2)0 = −
Xm6=n
|H(1)mn|2
ε(0)m − ε
(0)0
(5.65)
is always negative (the change in state always leads to a lower energy, consistent with thevariational principle), because the energy denominators are always positive. We can thuswrite ¯
ε(2)0
¯=Xm6=n
|H(1)mn|2
ε(0)m − ε
(0)0
≥ |H(1)mn|2
ε(0)m − ε
(0)0
(5.66)
where in the right-hand side we have used the fact that any single term in the sum is lessthan or equal to the total sum of all the positive defnitite terms therein. The maximalterm in the sum (which is usually one of the low-lying excited states closest in energy tothe ground state) can thus be used to provide a reasonable lower bound for the the secondorder shift in the ground state energy.
As an application of the techniques of nondegenerate perturbation theory weconsider the example of a harmonically bound electron to which a uniform electric fieldis applied. Thus, we take for our Hamiltonian
H = H0 + V (5.67)
where
H0 =P 2
2m+1
2mω2X2 (5.68)
is a simple one-dimensional harmonic oscillator describing the bound electron, and theperturbing field is described by the potential
V = −eEX = −fX. (5.69)
Perturbation Theory for Nondegenerate Levels 143
Our goal is to treat V as a small perturbation and calculate relevant corrections to theenergy levels and eigenstates in the presence of the applied electric field. To this end werecall the standard transformations
q =
rmω
~X p =
P√m~ω
(5.70)
a =1√2(q + ip) a+ =
1√2(q − ip) N = a+a (5.71)
that allow us to put the harmonic oscillator part of the problem in a simpler, dimensionlessform
H0 =
·N +
1
2
¸~ω V = −f
r~mω
q = −λq = − λ√2(a+ + a) (5.72)
where
λ = f
r~mω
(5.73)
is a (presumed small) measure of the strength of the applied field. With these definitions,the unperturbed states of H0 are the usual oscillator states |ni which obey
N |ni = n|ni H0|ni = (n+ 12)~ω|ni = ε(0)n |ni hn|n0i = δn,n0 . (5.74)
We also have the relations
a+|ni = √n+ 1|n+ 1i a|ni = √n|n− 1i (5.75)
in terms of which we readily determine that the first order energy shift due to the appliedfield
ε(1)n = hn|V |ni = −λ√2hn|(a+ + a)|ni
= − λ√2
©hn|a+|ni+ hn|a|niª = − λ√2
©√n+ 1hn|n+ 1i+√nhn|n− 1iª
= 0 (5.76)
vanishes due to the orthogonality of the unperturbed states. So the first order energyshift vanishes and we must go to second order to calculate the energy shift. Physicallythis vanishing of the first order energy shift occurs for the unperturbed states becausethey have equal weight on each side of the origin, and so the net change in energy due tothe linear applied potential vanishes. We can anticipate that the second order correctionwill cause a lowering of the energy as the electron displaces in the presence of the field,lowering its potential energy in the process. To see this we first calculate the first ordercorrection to the eigenstates. In the present problem we will denote by |ni the exacteigenstates of the system that are presumed to have an expansion
|ni = |ni+ λ|ni(1) + λ2|ni(2) + · · · (5.77)
in powers of the small parameter λ. The first order correction is given, according to theresults of the last section, by the expression
λ|ni(1) =Xm6=n
hm|V |niε(0)n − ε
(0)m
|mi = − λ√2
Xn0 6=n
hm| (a+ + a) |ni(n−m)~ω |mi. (5.78)
144 Approximation Methods for Stationary States
Letting a and a+ act to the right we find after a little calculation that, except for theground state, this reduces to a sum of just two terms
λ|ni(1) = − λ√2
·√n+ 1
−~ω |n+ 1i+√n
~ω|n− 1i
¸=
λ√2~ω
£√n+ 1|n+ 1i−√n|n− 1i¤ . (5.79)
Thus, to first order the exact eigenstates of H can be written
|ni = |ni+ λ√2~ω
£√n+ 1|n+ 1i−√n|n− 1i¤+O(λ2). (5.80)
Thus, the perturbation mixes only the states immediately above and below the unper-turbed level. Note that the corresponding expression for the ground state does not containthe second term in the above expression, i.e.,
|0i = |0i+ λ√2~ω
|1i+O(λ2). (5.81)
We now consider the second order energy shift
ε(2)n =Xm6=n
¯hm|V |ni
¯2ε(0)n − ε
(0)m
(5.82)
which will also (except for the ground state) contain just two terms:
ε(2)n =λ2
2
(|hn+ 1|a+|ni|2
−~ω +|hn− 1|a|ni|2
~ω
)
=λ2
2
½−n+ 1~ω
+n
~ω
¾= − λ2
2~ω= − f2
2mω2. (5.83)
Thus, to second order, we find
εn = ε(0)n −λ2
2~ω= − f2
2mω2. (5.84)
We note that for this particular problem the energy shift is the same for all states, thatis, all of the energies of the system are lowered by the same amount in the presence ofthe field. It turns out that the second order energy correction for this problem givesthe exact eigenenergies (even though the first order correction to the state does not givethe exact eigenstates). This result is physically inuitive, since it corresponds to the factthat a classical mass-spring system when hung in a gravitationl field simply stretches,or displaces, to a new equilibrium position, thereby lowering its potential energy, butcontinues to oscillate with the same frequency as it would if it were left unperturbed. Toestablish this result in the present context we perform a canonical transform to a new setof variables
q = q − λ
~ωp = p (5.85)
which has the position coordinate now centered at the new force center at q = λ/~ω. Thistransformation preserves the commutation relations
[q, p] = [q, p] = i (5.86)
Perturbation Theory for Nondegenerate Levels 145
and allows us to write the Hamiltonian in terms of the new coordinate and momenta inthe form
H =~ω2(q2 + p2)− λq
=~ω2
·q2 +
2λ
~ωq +
λ2
~2ω2+ p2
¸− λ
µq +
λ
~ω
¶=
~ω2(q2 + p2)− λ2
2~ω. (5.87)
We can now introduce operators
a =q + ip√2
a+ =q − ip√2
N = a+a
in the usual way, so that the Hamiltonian takes the form
H = (N +1
2)~ω − λ2
2~ω
of an oscillator of frequency ω lowered uniformly in energy by an amount λ2/2~ω. Ofcourse this oscillator has its equilibrium position at q = q − λ/~ω = 0 , i.e., shifted withrespect to the unperturbed oscillator, and its energy levels are in exact agreement withthose found to second order in the applied field using perturbation theory. In fact, thisdisplacement of the oscillator under the action of the field makes it clear that the exacteigenstates of the system satisfy the equation
hq|ni = φn(q) = φn(q − λ/~ω) = φn(q − ε),
where ε = λ/~ω, i.e., they are just the unperturbed oscillator states shifted along thex-axis, and centered at the new equilibrium position q = ε. The unitary operator whicheffects this transformation is the corresponding translation operator
T (ε) = e−ipε
which, in the position representation has the effect of displacing the wave function, i.e.,T (ε)ψ(q) = ψ(q − ε). Thus, we expect that the unperturbed and perturbed eigenstatesare related through the relation
|ni = T (ε)|ni = e−ipε|ni.
For small ε (or small λ) we can expand the exponential as T (ε) ' 1− ipε so that
|ni ' (1− ipε)|ni = |ni− ipε|ni.
Using the fact that −ipε = ε(a+ − a)/√2 and substituting back in the definition ofε = λ/~ω we recover the result
|ni ' |ni+ λ√2~ω
|n+ 1i− λ√2~ω
|n− 1i
that we obtained using the techniques of first order perturbation theory.
146 Approximation Methods for Stationary States
5.3 Perturbation Theory for Degenerate States
The expressions that we derived above for the first order correction to the ground state andthe second order correction to the energy are clearly inappropriate to situations in whichdegenerate or nearly-degenerate eigenstates are connected by the perturbation, since thecorresponding corrections all diverge as the spacing between the energy levels goes to zero.This divergence is an indication of the strength with which the perturbation tries to mixtogether states that are very-nearly degenerate (or exactly so), and suggest that we mightwish to treat differnyl those states that are known in advance to be very closely relatedin energy. In this section, therefore, we discuss the general approach taken to deal withproblems of this sort.
We assume, as before, that the Hamiltonian of interest can be separated into twoparts, which we now write in the simplied form
H = H0 + V (5.88)
where, since we will not be developing a systematic expansion in powers of the pertur-bation we have no need for the more complex notation used previously. We again seekthe exact eigenstates and eigenenergies of H, expressed as an expansion in eigenstates ofH0, the latter of which are assumed to be at least partially degenerate. We will denoteby |φn, τi an arbitrary othonormal basis of eigenstates of H0, where the index τ is in-cluded to distinguish between the different linearly independent basis states of H0 havingthe same unperturbed energy ε(0)n . The basis states
|φn, τi | τ = 1, · · · , Nn (5.89)
with fixed energy ε(0)n form a basis for an eigensubspace S(ε(0)n ) of H0 correspondingto that particular degenerate energy. The dimension Nn of this subspace is just thedegeneracy of the corresponding eigenvalue ε(0)n of H0. It is important to point out that,due to the degeneracy, our choice of the basis set |φn, τi is not unique; any unitarytransformation carried out within any one of the eigenspaces S(ε(0)n ) generates a new basis|χn, τi that can be used as readily as any other for expanding the exact eigenstates ofH. Any such basis set will satisfy the obvious eigenvalue, orthogonality, and completenessrelations
H0|φn, τi = ε(0)n |φn, τi hφn0 , τ 0|φn, τi = δn0,nδτ 0,τXn,τ
|φn, τihφn, τ | = 1
H0|χn, τi = ε(0)n |χn, τi hχn0 , τ 0|χn, τi = δn0,nδτ 0,τXn,τ
|χn, τihχn, τ | = 1.(5.90)
We now observe that the divergences that render the formulae of nondegenerateperturbation theory inapplicable really only arise if the perturbation actually connectsstates within each eigensubspace, i.e., if there exists non-zero matrix elements Vnτ ,n0τ 0 =hφn,τ |V |φn,τ 0i of the perturbation connecting basis states of the same energy. Thus, ourprevious formulae can, in fact be applied (at least to the level that we have developedthem), under two conceivable circumstances, one involving the diagonal matrix elementsof V and one involving the off-diagonal elements:
1. If the first order correction ε(1)n,τ = hφn, τ |V |φn, τi is distinct for all the basis states
|φn, τi in each eigenspace S(ε(0)n ), then the degeneracy is “lifted” in the first order ofthe perturbation. Provided the magnitude of this splitting of the energy levels by theperturbation is large compared to the matrix elements of V that connect these states
Perturbation Theory for Degenerate States 147
we can then simply “redefine” what we call the unperturbed and the perturbingpart of the Hamiltonian. In other words, although we orginally decomposed theHamiltonian in the form H = H0 + V , where, in a representation of eigenstates ofH0,
H0 =Xn,τ
|φn, τiε(0)n hφn, τ |
V =X
n,τ ;n0,τ 0|φn, τiVnτ ,n0τ 0hφn0 , τ 0|,
we can now include the diagonal part of V in a redefined H0 such that H = H0+ V ,but now,
H0 =Xn,τ
|φn, τi[ε(0)n + ε(1)n,τ ]hφn, τ |
V =X
n,τ 6=n0,τ 0|φn, τiVnτ ,n0τ 0hφn0 , τ 0|
where the perturbation V now has no diagonal matrix elements in this representa-tion. In this situation, states within S(ε(0)n ) are now no longer degenerate, so wecan proceed as before to apply the formulae of non-degenerate perturbation theory,with the energy denominators now including the first order shifts, so no divergencesoccur.
2. If, on the other hand, the off-diagonal part of the perturbation V just happens tovanish between all the basis states within a given eigensubspace S(ε(0)n ), then (atleast to second order) the problematic terms in the perturbation expansion neveractually arise; thus if the submatrix [V ]n representing the perturbation within thedegenerate subspace is diagonal, we can actually proceed as though there were nodegeneracy.
In passing we might comment regarding the first of these circumstances that theact of including the diagonal part of the perturbation V in a redefined H0 can always beperformed, even when it does not entirely lift the degeneracy. We may therefore assumewithout loss of generality in what follows that such an operation has already been carriedout, and hence that the perturbation has no diagonal components in the basis of interest.
Regarding the second circumstance mentioned above, it might be thought that,in actual practice, the vanishing of the off-diagoanl matrix elements of V within S(ε(0)n )would occur in so few circumstances that it hardly merits attention. To the contrary,there is a sense in which it always can be made to occur. To understand this comment,and in a the process reveal the basic technique that is generally employed for dealing withdegenerate states, we note that the off-diagonal matrix elements of the perturbation Vtaken between basis states in a given degenerate subspace of H0 depend upon which setof basis states of H0 we choose to begin with. If, e.g., we choose a set |φn, τi we getone set of matrix elements
Vnτ ;nτ 0 = hφn, τ |V |φn, τi,defining a certain submatrix [V ]n, while if we choose, instead, any other set |χn, τi weobtain a completely different set of matrix elements
Vnτ ;nτ 0 = hχn, τ |V |χn, τ 0i
148 Approximation Methods for Stationary States
defining a different submatrix [V ] representing the perturbation V within this degenerateeigensubspace.
Thus, as we would expect, the submatrix [V ]n or [V ]n representing the pertur-
bation within the eigenspace S(ε(0)n ) depends upon the particular basis set (|φn, τi or|χn, τi) that we chose to work in. In light of circumstance 2, above, the question thatarises is the following: under what circumstances can we find a representation of basis
states |χn, τi within S(ε(0)n ) for which the submatrixhVirepresenting the perturbation
in that subspace is strictly diagonal?Insofar as the perturbation V itself is presumed to be an observable (and thus
Hermitian), any submatrix [V ]n or [V ]n representing V within such a subspace mustitself be a Hermitian, and related to any of the other matrices representing V withinthis subspace by a unitary (sub)tranformation. For a finite Nn dimensional subspace,however, we know that it is always possible to find a representation that diagonalizes anyHermitian matrix. For each subspace we just have to go through the usual procedure offinding the roots εn,τ to the characteristic equation
det ([V ]n − ε) = 0
for theNn dimensional submatrix [V ]n that represents V within a given eigenspace S(ε(0)n ),
and then solve the resulting linear equations to find those combinations |χn, τi of theoriginal basis vectors |φn, τi that are also eigenvectors of the submatrix [V ]n. In thisnew representation, by construction, no elements of the new basis set |χn, τi havingthe same energy unperturbed energy are connected to one another by nonzero matrixelements of the perturbation. Moreover, the diagonalization of V within each eigenspaceS(ε
(0)n ) provides a new set of eigenvalues εn,τ (the roots of the characteristic equation
det [V − ε]n computed within the subspace) which will form the diagonal elements ofthe matrix representing V in this representation. These diagonal elements can then becombined with those ofH0 to obtain new unperturbed eigenenergies (correct to first order)
εn,τ = ε(0)n + εn,τ
that will themselves often at least partially lift the degeneracy. The particular states foundduring the diagonalization can then be chosen as a new set of zeroth order states withwhich to pursue higher order corrections, according to the techniques of nondegenerateperturbation theory. Thus, the basic result of degenerate perturbation theory is not anexplicit formula, as it is in the nondegenerate case. Rather it is a simple prescription:diagonalize the perturbation V within the degenerate subspaces of H0 to determine a newbasis for proceeding, if necessary, with the determination of higher order corrections usingstandard techniques.5.3.1 Application: Stark Effect of the n = 2 Level of Hydrogen
We consider as an application of the ideas developed above the splitting of the spectrallines observed in the absorbtion and emission spectra of the hydrogen atom when it isplaced in a uniform DC electric field, the so-called Stark effect. The relevant Hamiltoniancan be written in the expected form
H = H0 + V,
where
H =p2
2m− e
2
ris the usual one describing a single electron bound to the proton of a hydrogen atom, andthe perturbation
V = Fz = Fr cos θ
Perturbation Theory for Degenerate States 149
describes the constant force F = eE exerted on the electron by a uniform field orientedalong the negative z-axis. We initially take as our unperturbed states the standard boundeigenstates |n, l,mi of the hydrogen atom
H0|n, l,mi = ε(0)n |n, l,mi ε(0)n = − ε0n2
which have both a rotational and an accidental degeneracy of the energy levels. Thedegeneracy gn of the nth eigenenergy (or the dimension Nn of the associated eigenspaceSn) is given by the expression
gn = n2 =
n−1X`=0
(2`+ 1).
The first order correction to the energy of the state |n, l,mi due to the applied electricfield vanishes, i.e.,
ε(1)nlm = F hnlm|Z|nlmi = 0
reflecting the fact that the mean position of the electron in any of the standard hydrogenatom eigenstates is the origin. To use perturbation theory to find non-vanishing correc-tions to the energy due to the applied field we must handle the degeneracies. Consider,e.g., the four-fold degenerate n = 2 level, which is spanned by the four |nlmi states
|2, 0, 0i |2, 1, 0i |2, 1, 1i |2, 1,−1i.
Within the subspace S2 the submatrix representing H0 is, of course, diagonal
[H0] =
ε(0)2 0 0 0
0 ε(0)2 0 0
0 0 ε(0)2 0
0 0 0 ε(0)2
.To proceed we need to construct the matrix [V ] representing the perturbation within thissubspace. Thus we need to evaluate the matrix elements
h2, l,m|z|2, l0,m0i =Zd3r ψ∗2,`,mzψ2,`0,m0 .
But the perturbing operator is clearly just the z component of the vector operator ~R.According to the Wigner-Eckart theorem such an operator can only connect states havingthe same z-component of angular momentum, i.e., those for which m = m0. Thus, of thestates in the n = 2 manifold, the only nonzero matrix elements for this perturbation occurbetween the state |2, 0, 0i and the state |2, 1, 0i. Hence, within this subspace the matrixof interest has the form
[V ] =
0 η 0 0η∗ 0 0 00 0 0 00 0 0 0
,where η = h2, 0, 0|Fz|2, 1, 0i. This latter integral is readily evaluated in the position rep-resentation, using the known form
h~r|2, 0, 0i = ψ2,0,0(~r) =1√8a3
(2− ra)e−r/2aY 00 (θ,φ)
150 Approximation Methods for Stationary States
h~r|2, 1, 0i = ψ2,1,0(~r) =1√8a3
1√3
r
ae−r/2aY 01 (θ,φ)
of the hydrogen n = 2 wave functions. Substituting into the integral of interest we findafter a short calculation that
η =F
16a4
·Z ∞0
r4(2− ra)e−r/adr
¸ ·Z π
0
sin θ cos2 θdθ
¸= −3Fa.
Thus,
[V ] =
0 −3Fa 0 0
−3Fa 0 0 00 0 0 00 0 0 0
.Diagonalizing [V ] we set det(V − ε) = −ε2[ε2 − (3Fa)2] = 0 and find the eigenvalues
ε2,1,1 = ε2,1,−1 = 0ε2,+ = +3Fa
ε2,− = −3Fa
which we can add to the n = 2 hydrogenic energies to provide the first order corrections.Thus, to first order, the n = 2 eigenenergies in the presence of the field take the form
ε2,+ = ε(0)2 + 3Fa
ε2,− = ε(0)2 − 3Fa
ε2,1,1 = ε2,1,−1 = ε(0)2
which correspond, respectively, to new zeroth order states
|2,+i =|2, 0, 0i+ |2, 0, 1i√
2
|2,−i =|2, 0, 0i+ |2, 0, 1i√
2
|2, 1, 1i|2, 1,−1i.
Qualitatively we see that the four-fold degenerate n = 2 hydrogenic level is split by thefield into three separate levels, with the nondegenerate lower and higher energy statessplitting off linearlyt in the applied field from the remaining two-fold degenerate subspacecorresponding to the unperturbed energies. With these new basis states (in which ` isno longer a necessarily good quantum) one can, in principle, investigate higher ordercorrections to the energy.
Chapter 5ANGULAR MOMENTUM AND ROTATIONS
In classical mechanics the total angular momentum ~L of an isolated system about any…xed point is conserved. The existence of a conserved vector ~L associated with such asystem is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian)is invariant under rotations, i.e., if the coordinates and momenta of the entire system arerotated “rigidly” about some point, the energy of the system is unchanged and, moreimportantly, is the same function of the dynamical variables as it was before the rotation.Such a circumstance would not apply, e.g., to a system lying in an externally imposedgravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolatedsystem under rotations ultimately arises from the fact that, in the absence of external…elds of this sort, space is isotropic; it behaves the same way in all directions.
Not surprisingly, therefore, in quantum mechanics the individual Cartesian com-ponents Li of the total angular momentum operator ~L of an isolated system are alsoconstants of the motion. The di¤erent components of ~L are not, however, compatiblequantum observables. Indeed, as we will see the operators representing the componentsof angular momentum along di¤erent directions do not generally commute with one an-other. Thus, the vector operator ~L is not, strictly speaking, an observable, since it doesnot have a complete basis of eigenstates (which would have to be simultaneous eigenstatesof all of its non-commuting components). This lack of commutivity often seems, at …rstencounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlyingstructure of the three dimensional space in which we are immersed, and has its sourcein the fact that rotations in three dimensions about di¤erent axes do not commute withone another. Indeed, it is this lack of commutivity that imparts to angular momentumobservables their rich characteristic structure and makes them quite useful, e.g., in classi-fying the bound states of atomic, molecular, and nuclear systems containing one or moreparticles, and in decomposing the scattering states of such systems into components as-sociated with di¤erent angular momenta. Just as importantly, the existence of internal“spin” degrees of freedom, i.e., intrinsic angular momenta associated with the internalstructure of fundamental particles, provides additional motivation for the study of angu-lar momentum and to the general properties exhibited by dynamical quantum systemsunder rotations.
5.1 Orbital Angular Momentum of One or More Particles
The classical orbital angular momentum of a single particle about a given origin is givenby the cross product
~= ~r £ ~p (5.1)
of its position and momentum vectors. The total angular momentum of a system of suchstructureless point particles is then the vector sum
~L =X®
~® =
X®
~r® £ ~p® (5.2)
162 Angular Momentum and Rotations
of the individual angular momenta of the particles making up the collection. In quantummechanics, of course, dynamical variables are replaced by Hermitian operators, and so weare led to consider the vector operator
~= ~R£ ~P (5.3)
or its dimensionless counterpart
~l = ~R£ ~K;=~
~; (5.4)
either of which we will refer to as an angular momentum (i.e., we will, for the rest of thischapter, e¤ectively be working in a set of units for which ~ = 1). Now, a general vectoroperator ~B can always be de…ned in terms of its operator components fBx; By; Bzg alongany three orthogonal axes. The component of ~B along any other direction, de…ned, e.g.,by the unit vector u; is then the operator ~B ¢ u = Bxux +Byuy +Bzuz. So it is with theoperator ~l; whose components are, by de…nition, the operators
lx = Y Kz ¡ ZKy ly = ZKx ¡XKz lz = XKy ¡ Y Kx: (5.5)
The components of the cross product can also be written in a more compact form
li =Xj;k
"ijkXjKk (5.6)
in terms of the Levi-Civita symbol
"ijk =
8<: 1 if ijk is an even permutation of 123¡1 if ijk is an odd permutation of 1230 otherwise
. (5.7)
Although the normal product of two Hermitian operators is itself Hermitian ifand only if they commute, this familiar rule does not extend to the cross product of twovector operators. Indeed, even though ~R and ~K do not commute, their cross product ~l isreadily shown to be Hermitian. From (5.6),
l+i =Xj;k
"ijkK+k X
+j =
Xj;k
"ijkKkXj =Xj;k
"ijkXjKk = li; (5.8)
where we have used the fact the components of ~R and ~K are Hermitian and that, since"ijk = 0 if k = j; only commuting components of ~R and ~K appear in each term of thecross product: It is also useful to de…ne the scalar operator
l2 = ~l ¢~l = l2x + l2y + l2z (5.9)
which, being the sum of the squares of Hermitian operators, is itself both Hermitian andpositive.
So the components of ~l; like those of the vector operators ~R and ~P ; are Hermitian.We will assume that they are also observables. Unlike the components of ~R and ~P ;however, the components of ~l along di¤erent directions do not commute with each other.This is readily established; e.g.,
[lx; ly] = [YKz ¡ ZKy; ZKx ¡KzX]= YKx [Kz; Z] +KyX [Z;Kz]
= i (XKy ¡ YKx) = ilz:
Orbital Angular Momentum of One or More Particles 163
The other two commutators are obtained in a similar fashion, or by a cyclic permutationof x; y; and z; giving
[lx; ly] = ilz [ly; lz] = ilx [lz; lx] = ily; (5.10)
which can be written more compactly using the Levi-Civita symbol in either of two ways,
[li; lj ] = iXk
"ijklk; (5.11)
or Xi;j
"ijklilj = ilk;
the latter of which is, component-by-component, equivalent to the vector relation
~l £~l = i~l: (5.12)
These can also be used to derive the following generalizationh~l ¢ a;~l ¢ b
i= i~l ¢
³a£ b
´(5.13)
involving the components of ~l along arbitrary directions a and b.It is also straightforward to compute the commutation relations between the com-
ponents of ~l and l2, i.e.,£lj ; l
2¤=
Xi
£lj ; l
2i
¤=Xi
li [lj; li] +Xi
[lj ; li] li
= iXi;k
("ijklilk + "ijklkli) = iXi;k
("ijklilk + "kjililk)
= iXi;k
"ijk(lilk ¡ lilk) = 0 (5.14)
where in the second line we have switched summation indices in the second sum and thenused the fact that "kji = ¡"ijk: Thus each component of ~l commutes with l2: We writeh
~l; l2i= 0 [li; lj ] = i
Xk
"ijklk: (5.15)
The same commutation relations are also easily shown to apply to the operatorrepresenting the total orbital angular momentum ~L of a system of particles. For such asystem, the state space of which is the direct product of the state spaces for each particle,the operators for one particle automatically commute with those of any other, so that
[Li; Lj ] =X®;¯
[li;®; lj;¯] = iXk
"ijkX®;¯
±®;¯ lk;® = iXk
"ijkX®
lk;®
= iXk
"ijkLk (5.16)
Similarly, from these commutation relations for the components of ~L, it can be shownthat
£Li; L
2¤= 0 using the same proof as above for ~l. Thus, for each particle, and for
the total orbital angular momentum itself, we have the same characteristic commutationrelations
[Li; Lj ] = iXk
"ijkLkh~L;L2
i= 0: (5.17)
164 Angular Momentum and Rotations
As we will see, these commutation relations determine to a very large extent the allowedspectrum and structure of the eigenstates of angular momentum. It is convenient to adoptthe viewpoint, therefore, that any vector operator obeying these characteristic commuta-tion relations represents an angular momentum of some sort. We thus generally say thatan arbitrary vector operator ~J is an angular momentum if its Cartesian components areobservables obeying the following characteristic commutation relations
[Ji; Jj ] = iXk
"ijkJk
h~J; J2
i= 0: (5.18)
It is actually possible to go considerably further than this. It can be shown,under very general circumstances, that for every quantum system there must exist avector operator ~J obeying the commutation relations (5.18), the components of whichcharacterize the way that the quantum system transforms under rotations. This vectoroperator ~J can usually, in such circumstances, be taken as a de…nition of the total angularmomentum of the associated system. Our immediate goals, therefore, are twofold. Firstwe will explore this underlying relationship that exists between rotations and the angularmomentum of a physical system. Then, afterwards, we will return to the commutationrelations (5.18), and use them to determine the allowed spectrum and the structure of theeigenstates of arbitrary angular momentum observables.
5.2 Rotation of Physical Systems
A rotation R of a physical system is a distance preserving mapping of R3 onto itselfthat leaves a single point O; and the handedness of coordinate systems invariant. Thisde…nition excludes, e.g., re‡ections and other “improper” transformations, which alwaysinvert coordinate systems. There are two di¤erent, but essentially equivalent ways ofmathematically describing rotations. An active rotation of a physical system is one inwhich all position and velocity vectors of particles in the system are rotated about the…xed point O; while the coordinate system used to describe the system is left unchanged.A passive rotation, by contrast, is one in which the coordinate axes are rotated, butthe physical vectors of the system are left alone. In either case the result, generally, isa change in the Cartesian components of any vector in the system with respect to thecoordinate axes used to represent them. It is important to note, however, that a clockwiseactive rotation of a physical system about a given axis is equivalent in terms of the changeit produces on the coordinates of a vector to a counterclockwise passive rotation aboutthe same axis.
There are also two di¤erent methods commonly adopted for indicating speci…crotations, each requiring three independent parameters. One method speci…es particularrotations through the use of the so-called Euler angles introduced in the study of rigidbodies. Thus, e.g., R(®; ¯; °) would indicate the rotation equivalent to the three separaterotations de…ned by the Euler angles (®; ¯; °):
Alternatively, we can indicate a rotation by choosing a speci…c rotation axis,described by a unit vector u (de…ned, e.g., through its polar angles µ and Á), and arotation angle ®: Thus, a rotation about u through an angle ® (positive or negative,according to the right-hand-rule applied to u) would be written Ru(®): We will, in whatfollows, make more use of this latter approach than we will of the Euler angles.
Independent of their means of speci…cation, the rotations about a speci…ed pointO in three dimensions form a group, referred to as the three-dimensional rotation group.Recall that a set G of elements R1; R2; ¢ ¢ ¢ ; that is closed under an associative binaryoperation,
RiRj = Rk 2 G for all Ri; Rj 2 G; (5.19)
Rotation of Physical Systems 165
is said to form a group if (i) there exists inG an identity element 1 such that R1 = 1R = Rfor all R in G and (ii) there is in G; for each R; an inverse element R¡1; such thatRR¡1=R¡1R = 1.
For the rotation group fRu(®)g the product of any two rotations is just therotation obtained by performing each rotation in sequence, i.e., Ru(®)Ru0(®
0) correspondsto a rotation of the physical system through an angle ®0 about u0; followed by a rotationthrough ® about u: The identity rotation corresponds to the limiting case of a rotation of® = 0 about any axis (i.e., the identity mapping). The inverse of Ru(®) is the rotation
R¡1u (®) = Ru(¡®) =R¡u(®); (5.20)
that rotates the system in the opposite direction about the same axis.It is readily veri…ed that, in three dimensions, the product of two rotations gen-
erally depends upon the order in which they are taken. That is, in most cases,
Ru(®)Ru0(®0) 6= Ru0(®
0)Ru(®): (5.21)
The rotation group, therefore, is said to be a noncommutative or non-Abelian group.There are, however, certain subsets of the rotation group that form commutative
subgroups (subsets of the original group that are themselves closed under the samebinary operation). For example, the set of rotations fRu(®) j …xed ug about any single…xed axis forms an Abelian subgroup of the 3D rotation group, since the product of tworotations in the plane perpendicular to u corresponds to a single rotation in that planethrough an angle equal to the (commutative) sum of the individual rotation angles,
Ru(®)Ru(¯) = Ru(®+ ¯) =Ru(¯)Ru(®): (5.22)
The subgroups of this type are all isomorphic to one another. Each one forms a realizationof what is referred to for obvious reasons as the two dimensional rotation group.
Another commutative subgroup comprises the set of in…nitesimal rotations.A rotation Ru(±®) is said to be in…nitesimal if the associated rotation angle ±® is anin…nitesimal (it being understood that quantities of order ±2® are always to be neglectedwith respect to quantities of order ±®). The e¤ect of an in…nitesimal rotation on a physicalquantity of the system is to change it, at most, by an in…nitesimal amount. The generalproperties of such rotations are perhaps most easily demonstrated by considering theire¤ect on normal vectors of R3.
The e¤ect of an arbitrary rotation R on a vector ~v of R3 is to transform it intoa new vector
~v0 = R [~v] : (5.23)
Because rotations preserves the relative orientations and lengths of all vectors in thesystem, it also preserves the basic linear relationships of the vector space itself, i.e.,
R [~v1 + ~v2] =R [~v1] +R [~v2] : (5.24)
Thus, the e¤ect of any rotation R on vectors in the R3 can be described through theaction of an associated linear operator AR; such that
R [~v] = ~v0 = AR~v: (5.25)
This linear relationship can be expressed in any Cartesian coordinate system in componentform
v0i =Xj
Aijvj (5.26)
166 Angular Momentum and Rotations
A systematic study of rotations reveals that the 3 £ 3 matrix A representing the linearoperator AR must be real, orthogonal, and unimodular, i.e.
Aij = A¤ij ATA = AAT = 1 det(A) = 1: (5.27)
We will denote by Au(®) the linear operator (or any matrix representation thereof, de-pending upon the context) representing the rotation Ru(®). The rotations Ru(®) and theorthogonal, unimodular matrices Au(®) representing their e¤ect on vectors with respectto a given coordinate system are in a one-to-one correspondence. We say, therefore, thatthe set of matrices fAu(®)g forms a representation of the 3D rotation group. The groupformed by the matrices themselves is referred to as SO3, which indicates the group of “spe-cial” orthogonal 3£ 3 matrices (special in that it excludes those orthogonal matrices thathave determinant of ¡1; i.e., it excludes re‡ections and other improper transformations).In this group, the matrix representing the identity rotation is, of course, the identitymatrix, while rotations about the three Cartesian axes are e¤ected by the matrices
Ax(µ) =
0@ 1 0 00 cos µ ¡ sin µ0 sin µ cos µ
1A Ay(µ) =
0@ cos µ 0 sin µ0 1 0
¡ sin µ 0 cos µ
1A
Az(µ) =
0@ cos µ ¡ sin µ 0sin µ cos µ 00 0 1
1A (5.28)
Now it is intuitively clear that the matrix associated with an in…nitesimal rotationbarely changes any vector that it acts upon and, as a result, di¤ers from the identity matrixby an in…nitesimal amount, i.e.,
Au(±®) = 1+ ±®Mu (5.29)
where Mu is describes a linear transformation that depends upon the rotation axis u butis independent of the in…nitesimal rotation angle ±®: The easily computed inverse
A¡1u (±®) = Au(¡±®) = 1¡ ±®Mu (5.30)
and the orthogonality of rotation matrices
A¡1u (±®) = ATu (±®) = 1+ ±®M
Tu (5.31)
leads to the requirement that the matrix
Mu = ¡MTu (5.32)
be real and antisymmetric. Thus, under such an in…nitesimal rotation, a vector ~v is takenonto the vector
~v0 = ~v + ±®Mu~v: (5.33)
Rotation of Physical Systems 167
dv = v δα sin θ
u
v’
δα
θ
v
Figure 1 Under an in…nitesimal rotation Ru(±®); the change d~v = ~v0 ¡ ~v in a vector ~v isperpendicular to both u and ~v; and has magnitude jd~vj = j~vj±® sin µ.
But an equivalent description of such an in…nitesimal transformation on a vectorcan be determined through simple geometrical arguments. The vector ~v0 obtained byrotating the vector ~v about u through an in…nitesimal angle ±® is easily veri…ed from Fig.(1) to be given by the expression
~v0 = ~v + ±® (u£ ~v) (5.34)
or, in component form
v0i = vi + ±®Xj;k
"ijkujvk (5.35)
A straightforward comparison of (5.33) and (5.34) reveals that, for these to be consistent,the matrix Mu must have matrix elements of the form Mik =
Pj "ijkuj ; i.e.,
Mu =
0@ 0 ¡uz uyuz 0 ¡ux¡uy ux 0
1A ; (5.36)
where ux; uy; and uz are the components (i.e., direction cosines) of the unit vector u. Notethat we can write (5.36) in the form
Mu =Xi
uiMi = uxMx + uyMy + uzMz (5.37)
where the three matricesMi that characterize rotations about the three di¤erent Cartesian
168 Angular Momentum and Rotations
axes are given by
Mx =
0@ 0 0 00 0 ¡10 1 0
1A My =
0@ 0 0 10 0 0¡1 0 0
1A Mz =
0@ 0 ¡1 01 0 00 0 0
1A : (5.38)
Returning to the point that motivated our discussion of in…nitesimal rotations,we note that
Au(±®)Au0(±®0) = (1+ ±®Mu) (1+ ±®
0Mu0)
= 1+ ±®Mu + ±®0Mu0 = Au0(±®)Au(±®
0); (5.39)
which shows that, to lowest order, a product of in…nitesimal rotations always commutes.This last expression also reveals that in…nitesimal rotations have a particularly simplecombination law, i.e., to multiply two or more in…nitesimal rotations simply add up theparts corresponding to the deviation of each one from the identity matrix. This rule,and the structure (5.37) of the matrices Mu implies the following important theorem:an in…nitesimal rotation Au(±®) about an arbitrary axis u can always be built up as aproduct of three in…nitesimal rotations about any three orthogonal axes, i.e.,
Au(±®) = 1+ ±®Mu
= 1+ ±®uxMx + ±®uyMy + ±®uzMz
which implies thatAu(±®) = Ax(ux±®)Ay(uy±®)Az(uz±®): (5.40)
Since this last property only involves products, it must be a group property associated withthe group SO3 of rotation matrices fAu(®)g ; i.e., a property shared by the in…nitesimalrotations that they represent, i.e.,
Ru(±®) = Rx(ux±®)Ry(uy±®)Rz(uz±®): (5.41)
We will use this group relation associated with in…nitesimal rotations in determining theire¤ect on quantum mechanical systems.
5.3 Rotations in Quantum Mechanics
Any quantum system, no matter how complicated, can be characterized by a set of ob-servables and by a state vector jÃi; which is an element of an associated Hilbert space.A rotation performed on a quantum mechanical system will generally result in a trans-formation of the state vector and to a similar transformation of the observables of thesystem. To make this a bit more concrete, it is useful to imagine an experiment set upon a rotatable table. The quantum system to be experimentally interrogated is describedby some initial suitably-normalized state vector jÃi: The experimental apparatus mightbe arranged to measure, e.g., the component of the momentum of the system along aparticular direction. Imagine, now, that the table containing both the system and theexperimental apparatus is rotated about a vertical axis in such a way that the quantumsystem “moves” rigidly with the table (i.e., so that an observer sitting on the table coulddistinguish no change in the system). After such a rotation, the system will generallybe in a new state jÃ0i; normalized in the same way as it was before the rotation. More-over, the apparatus that has rotated with the table will now be set up to measure themomentum along a di¤erent direction, as measured by a set of coordinate axes …xed inthe laboratory..
Rotations in Quantum Mechanics 169
Such a transformation clearly describes a mapping of the quantum mechanicalstate space onto itself in a way that preserves the relationships of vectors in that space,i.e., it describes a unitary transformation. Not surprisingly, therefore, the e¤ect of anyrotationR on a quantum system can quite generally be characterized by a unitary operatorUR; i.e.,
jÃ0i = R [jÃi] = URjÃi: (5.42)
Moreover, the transformation experienced under such a rotation by observables of thesystem must have the property that the mean value and statistical distribution of anobservable Q taken with respect to the original state jÃi will be the same as the meanvalue and distribution of the rotated observable Q0 = R [Q] taken with respect to therotated state jÃ0i; i.e.,
hÃjQjÃi = hÃ0jQ0jÃ0i = hÃjU+RQ0URjÃi (5.43)
From (5.43) we deduce the relationship
Q0 = R [Q] = URQU+R : (5.44)
Thus, the observable Q0 is obtained through a unitary transformation of the unrotatedobservable Q using the same unitary operator that is needed to describe the change in thestate vector. Consistent with our previous notation, we will denote by Uu(®) the unitarytransformation describing the e¤ect on a quantum system of a rotation Ru(®) about uthrough angle ®.
Just as the 3 £ 3 matrices fAu(®)g form a representation of the rotation groupfRu(®)g, so do the set of unitary operators fUu(®)g and so also do the set of matricesrepresenting these operators with respect to any given ONB for the state space. Also,as with the case of normal vectors in R3; an in…nitesimal rotation on a quantum systemwill produce an in…nitesimal change in the state vector jÃi: Thus, the unitary operatorUu(±®) describing such an in…nitesimal rotation will di¤er from the identity operator byan in…nitesimal, i.e.,
Uu(±®) = 1+ ±® Mu (5.45)
where Mu is now a linear operator, de…ned not on R3 but on the Hilbert space of thequantum system under consideration, that depends on u but is independent of ±®. Similarto our previous calculation, the easily computed inverse
U¡1u (±®) = Uu(¡±®) = 1¡ ±® Mu (5.46)
and the unitarity of these operators (U¡1 = U+) leads to the result that, now, Mu =¡M+
u is anti-Hermitian. There exists, therefore, for each quantum system, an Hermitianoperator Ju = iMu; such that
Uu(±®) = 1¡ i±®Ju: (5.47)
The Hermitian operator Ju is referred to as the generator of in…nitesimal rotationsabout the axis u: Evidently, there is a di¤erent operator Ju characterizing rotations abouteach direction in space. Fortunately, as it turns out, all of these di¤erent operators Jucan be expressed as a simple combination of any three operators Jx; Jy; and Jz describingrotations about a given set of coordinate axes. This economy of expression arises from thecombination rule (5.41) obeyed by in…nitesimal rotations, which implies a correspondingrule
Uu(±®) = Ux(ux±®)Uy(uy±®)Uz(uz±®)
170 Angular Momentum and Rotations
for the unitary operators that represent them in Hilbert space. Using (5.47), this funda-mental relation implies that
Uu(±®) = (1¡ i±®uxJx)(1¡ i±®uyJy)(1¡ i±®uzJz)= 1¡i±® (uxJx + uyJy + iuzJz): (5.48)
Implicit in the form of Eq. (5.48) is the existence of a vector operator ~J , with Hermitiancomponents Jx; Jy; and Jz that generate in…nitesimal rotations about the correspondingcoordinate axes, and in terms of which an arbitrary in…nitesimal rotation can be expressedin the form
Uu(±®) = 1¡ i±® ~J ¢ u = 1¡ i±®Ju (5.49)
where Ju = ~J ¢ u now represents the component of the vector operator ~J along u.From this form that we have deduced for the unitary operators representing in…n-
itesimal rotations we can now construct the operators representing …nite rotations. Sincerotations about a …xed axis form a commutative subgroup, we can write
Uu(®+ ±®) = Uu(±®)Uu(®) = (1¡ i±®Ju)Uu(®) (5.50)
which implies that
dUu(®)
d®= lim±®!0
Uu(®+ ±®)¡ Uu(®)±®
= ¡iJuUu(®): (5.51)
The solution to this equation, subject to the obvious boundary condition Uu(0) = 1; isthe unitary rotation operator
Uu(®) = exp (¡i®Ju) = exp³¡i® ~J ¢ ~u
´: (5.52)
We have shown, therefore, that a description of the behavior of a quantum sys-tem under rotations leads automatically to the identi…cation of a vector operator ~J; whosecomponents act as generators of in…nitesimal rotations and the exponential of which gen-erates the unitary operators necessary to describe more general rotations of arbitraryquantum mechanical systems. It is convenient to adopt the point of view that the vectoroperator ~J whose existence we have deduced represents, by de…nition, the total angu-lar momentum of the associated system. We will postpone until later a discussion ofhow angular momentum operators for particular systems are actually identi…ed and con-structed. In the meantime, however, to show that this point of view is at least consistentwe must demonstrate that the components of ~J satisfy the characteristic commutationrelations (5.18) that are, in fact, obeyed by the operators representing the orbital angularmomentum of a system of one or more particles.
5.4 Commutation Relations for Scalar and Vector Operators
The analysis of the last section shows that for a general quantum system there exists avector operator ~J; to be identi…ed with the angular momentum of the system, that isessential for describing the e¤ect of rotations on the state vector jÃi and its observablesQ. Indeed, the results of the last section imply that a rotation Ru(®) of the physicalsystem will take an arbitrary observable Q onto a generally di¤erent observable
Q0 = Uu(®)QU+u (®) = e¡i®JuQei®Ju : (5.53)
For in…nitesimal rotations Uu(±®), this transformation law takes the form
Q0 = (1¡ i±® Ju)Q(1+ i±®Ju) (5.54)
Commutation Relations for Scalar and Vector Operators 171
which, to lowest nontrivial order, implies that
Q0 = Q¡ i±® [Ju;Q] : (5.55)
Now, as in classical mechanics, it is possible to classify certain types of observables ofthe system according to the manner in which they transform under rotations. Thus, anobservable Q is referred to as a scalar with respect to rotations if
Q0 = Q; (5.56)
for all R: For this to be true for arbitrary rotations, we must have, from (5.53), that
Q0 = URQU+R = Q; (5.57)
which implies that URQ = QUR; or
[Q;UR] = 0: (5.58)
Thus, for Q to be a scalar it must commute with the complete set of rotation operators forthe space. A somewhat simpler expression can be obtained by considering the in…nitesimalrotations, where from (5.55) and (5.56) we see that the condition for Q to be a scalarreduces to the requirement that
[Ju; Q] = 0; (5.59)
for all components Ju; which implies thath~J;Q
i= 0: (5.60)
Thus, by de…nition, any observable that commutes with the total angular momentum ofthe system is a scalar with respect to rotations.
A collection of three operators Vx; Vy; and Vz can be viewed as forming the com-ponents of a vector operator ~V if the component of ~V along an arbitrary direction a isVa = ~V ¢ a = P
i Viai: By construction, therefore, the operator ~J is a vector operator,since its component along any direction is a linear combination of its three Cartesiancomponents with coe¢cients that are, indeed, just the associated direction cosines. Now,after undergoing a rotation R; a device initially setup to measure the component Va of avector operator ~V along the direction a will now measure the component of ~V along therotated direction
a0 = ARa; (5.61)
where AR is the orthogonal matrix associated with the rotation R: Thus, we can write
R [Va] = URVaU+R = UR(
~V ¢ a)U+R = ~V ¢ a0 = Va0 : (5.62)
Again considering in…nitesimal rotations Uu(±®); and applying (5.55), this reduces to therelation
~V ¢ a0 = ~V ¢ a¡ i±®h~J ¢ u; ~V ¢ a
i: (5.63)
But we also know that, as in (5.34), an in…nitesimal rotation Au(±®) about u takes thevector a onto the vector
a0 = (1+ ±®Mu) a = a+ ±® (u£ a) : (5.64)
Consistency of (5.63) and (5.64) requires that
~V ¢ a0 = ~V ¢ a+ ±® ~V ¢ (u£ a) = ~V ¢ a¡ i±®h~J ¢ u; ~V ¢ a
i(5.65)
172 Angular Momentum and Rotations
i.e., that h~J ¢ u; ~V ¢ a
i= i~V ¢ (u£ a): (5.66)
Taking u and a along the ith and jth Cartesian axes, respectively, this latter relation canbe written in the form
[Ji; Vj ] = iXk
"ijkVk; (5.67)
or more speci…cally
[Jx; Vy] = iVz [Jy; Vz] = iVx [Jz; Vx] = iVy (5.68)
which shows that the components of any vector operator of a quantum system obey com-mutation relations with the components of the angular momentum that are very similarto those derived earlier for the operators associated with the orbital angular momentum,itself. Indeed, since the operator ~J is a vector operator with respect to rotations, it mustalso obey these same commutation relations, i.e.,
[Ji; Jj ] =Xk
i"ijkJk: (5.69)
Thus, our identi…cation of the operator ~J identi…ed above as the total angular momentumof the quantum system is entirely consistent with our earlier de…nition, in which weidenti…ed as an angular momentum any vector operator whose components obey thecharacteristic commutation relations (5.18).
5.5 Relation to Orbital Angular Momentum
To make some of the ideas introduced above a bit more concrete, we show how thegenerator of rotations ~J relates to the usual de…nition of angular momentum for, e.g., asingle spinless particle. This is most easily done by working in the position representation.For example, let Ã(~r) = h~rjÃi be the wave function associated with an arbitrary statejÃi of a single spinless particle. Under a rotation R; the ket jÃi is taken onto a newket jÃ0i = URjÃi described by a di¤erent wave function Ã0(~r) = h~rjÃ0i: The new wavefunction Ã0, obtained from the original by rotation, has the property that the value of theunrotated wavefunction à at the point ~r must be the same as the value of the rotatedwave function Ã0 at the rotated point ~r0 = AR~r: This relationship can be written in severalways, e.g.,
Ã(~r) = Ã0(~r0) = Ã0(AR~r) (5.70)
which can be evaluated at the point A¡1R ~r to obtain
Ã0(~r) = Ã(A¡1R ~r): (5.71)
Suppose that in (5.71), the rotation AR = Au(±®) represents an in…nitesimal rotationabout the axis u through and angle ±®, for which
AR~r = ~r + ±®(u£ ~r): (5.72)
The inverse rotation A¡1R is then given by
A¡1R ~r = ~r ¡ ±® (u£ ~r) : (5.73)
Thus, under such a rotation, we can write
Ã0(~r) = Ã(A¡1R ~r) = Ã [~r ¡ ±® (u£ ~r)]= Ã (~r)¡ ±® (u£ ~r) ¢ ~rÃ(~r) (5.74)
Relation to Orbital Angular Momentum 173
where we have expanded à [~r ¡ ±® (u£ ~r)] about the point ~r; retaining …rst order in…ni-tesimals. We can use the easily-proven identity
(u£ ~r) ¢ ~rà = u ¢ (~r £ ~r)Ãwhich allows us to write
Ã0(~r) = Ã (~r)¡ ±® u ¢ (~r £ ~r)Ã(~r)
= à (~r)¡ i±® u ¢Ã~r £
~ri
!Ã(~r) (5.75)
In Dirac notation this is equivalent to the relation
h~rjÃ0i = h~rjUu(±®)jÃi = h~rj³1¡ i±® u ¢ ~ jÃi (5.76)
where~= ~R£ ~K: (5.77)
Thus, we identify the vector operator ~J for a single spinless particle with the orbitalangular momentum operator ~: This allows us to write a general rotation operator forsuch a particle in the form
Uu(®) = exp³¡i®~ ¢ u
´: (5.78)
Thus the components of ~ form the generators of in…nitesimal rotations.Now the state space for a collection of such particles can be considered the direct or
tensor product of the state spaces associated with each one. Since operators from di¤erentspaces commute with each other, the unitary operator U (1)R that describes rotations of oneparticle will commute with those of another. It is not di¢cult to see that under thesecircumstances the operator that rotates the entire state vector jÃi is the product of therotation operators for each particle. Suppose, e.g., that jÃi is a direct product state, i.e.,
jÃi = jÃ1; Ã2; ¢ ¢ ¢ ; ÃNi:Under a rotation R; the state vector jÃi is taken onto the state vector
jÃ0i = jÃ01; Ã02; ¢ ¢ ¢ ; Ã0Ni= U
(1)R jÃ01iU (2)R jÃ02i ¢ ¢ ¢U (N)R jÃ0Ni
= U(1)R U
(2)R ¢ ¢ ¢U(N)R jÃ1; Ã2; ¢ ¢ ¢ ; ÃNi
= URjÃiwhere
UR = U(1)R U
(2)R ¢ ¢ ¢U (N)R
is a product of rotation operators for each part of the space, all corresponding to the samerotation R: Because these individual operators can all be written in the same form, i.e.,
U(¯)R = exp
³¡i®~ ¢ u
´;
where ~ it the orbital angular momentum for particle ¯, it follows that the total rotationoperator for the space takes the form
UR =Y¯
exp³¡i®~ ¢ u
´= exp
³¡i®~L ¢ u
´
174 Angular Momentum and Rotations
where~L =
X¯
~ :
Thus, we are led naturally to the point of view that the generator of rotations for thewhole system is the sum of the generators for each part thereof, hence for a collection ofspinless particles the total angular momentum ~J coincides with the total orbital angularmomentum ~L; as we would expect.
Clearly, the generators for any composite system formed from the direct product ofother subsystems is always the sum of the generators for each subsystem being combined.That is, for a general direct product space in which the operators ~J1; ~J2; ¢ ¢ ¢ ~JN are thevector operators whose components are the generators of rotations for each subspace, thecorresponding generators of rotation for the combined space is obtained as a sum
~J =NX®=1
~J®
of those for each space, and the total rotation operator takes the form
Uu(®) = exp³¡i® ~J ¢ u
´:
For particles with spin, the individual particle spaces can themselves be considered directproducts of a spatial part and a spin part. Thus for a single particle of spin ~S the generatorof rotations are the components of the vector operator ~J = ~L+ ~S; where ~L takes care ofrotations on the spatial part of the state and ~S does the same for the spin part.
5.6 Eigenstates and Eigenvalues of Angular Momentum Operators
Having explored the relationship between rotations and angular momenta, we now under-take a systematic study of the eigenstates and eigenvalues of a vector operator ~J obeyingangular momentum commutation relations of the type that we have derived. As we willsee, the process for obtaining this information is very similar to that used to determinethe spectrum of the eigenstates of the harmonic oscillator. We consider, therefore, anarbitrary angular momentum operator ~J whose components satisfy the relations
[Ji; Jj ] = iXk
"ijkJkh~J; J2
i= 0: (5.79)
We note, as we did for the orbital angular momentum ~L; that, since the componentsJi do not commute with one another, ~J cannot possess an ONB of eigenstates, i.e.,states which are simultaneous eigenstates of all three of its operator components. In fact,one can show that the only possible eigenstates of ~J are those for which the angularmomentum is identically zero (an s-state in the language of spectroscopy). Nonetheless,since, according to (5.79), each component of ~J commutes with J2; it is possible to…nd an ONB of eigenstates common to J2 and to the component of ~J along any chosendirection. Usually the component of ~J along the z-axis is chosen, because of the simpleform taken by the di¤erential operator representing that component of orbital angularmomentum in spherical coordinates. Note, however, that due to the cyclical nature of thecommutation relations, anything deduced about the spectrum and eigenstates of J2 andJz must also apply to the eigenstates common to J2 and to any other component of ~J:Thus the spectrum of Jz must be the same as that of Jx; Jy; or Ju = ~J ¢ u:
We note also, that, as with l2; the operator J2 =Pi J
2i is Hermitian and positive
de…nite and thus its eigenvalues must be greater than or equal to zero. For the moment,
Eigenstates and Eigenvalues of Angular Momentum Operators 175
we will ignore other quantum numbers and simply denote a common eigenstate of J2 andJz as jj;mi; where, by de…nition,
Jzjj;mi = mjj;mi (5.80)
J2jj;mi = j(j + 1)jj;mi: (5.81)
which implies that m is the associated eigenvalue of the operator Jz; while the label j isintended to simply identify the corresponding eigenvalue j(j + 1) of J2: The justi…cationfor writing the eigenvalues of J2 in this fashion is only that it simpli…es the algebra andthe …nal results obtained. At this point, there is no obvious harm in expressing thingsin this fashion since for real values of j the corresponding values of j(j + 1) include allvalues between 0 and 1; and so any possible eigenvalue of J2 can be represented in theform j(j + 1) for some value of j. Moreover, it is easy to show that any non-negativevalue of j(j +1) can be obtained using a value of j which is itself non-negative. Withoutloss of generality, therefore, we assume that j ¸ 0:We will also, in the interest of brevity,refer to a vector jj;mi satisfying the eigenvalue equations (5.80) and (5.81) as a “vectorof angular momentum (j;m)”.
To proceed further, it is convenient to introduce the operator
J+ = Jx + iJy (5.82)
formed from the components of ~J along the x and y axes. The adjoint of J+ is the operator
J¡ = Jx ¡ iJy; (5.83)
in terms of which we can express the original operators
Jx =1
2(J+ + J¡) Jy =
i
2(J¡ ¡ J+) : (5.84)
Thus, in determining the spectrum and common eigenstates of J2 and Jz ,we will …ndit convenient to work with the set of operators
©J+; J¡; J2; Jz
ªrather than the set©
Jx; Jy; Jz; J2ª: In the process, we will require commutation relations for the opera-
tors in this new set. We note …rst that J§; being a linear combination of Jx and Jy; mustby (5.79) commute with J2: The commutator of J§ with Jz is also readily established;we …nd that
[Jz; J§] = [Jz; Jx]§ i [Jz; Jy] = iJy § Jx (5.85)
or[Jz; J§] = §J§: (5.86)
Similarly, the commutator of J+ and J¡ is
[J+; J¡] = [Jx; Jx] + [iJy; Jx]¡ [Jx; iJy]¡ [iJy; iJy]= 2Jz: (5.87)
Thus the commutation relations of interest take the form
[Jz; J§] = §J§ [J+; J¡] = 2Jz£J2; J§
¤= 0 =
£J2; Jz
¤: (5.88)
It will also be necessary in what follows to express the operator J2 in terms of the new“components” fJ+; J¡; Jzg rather than the old components fJx; Jy; Jzg : To this end wenote that J2 ¡ J2z = J2x + J2y ; and so
J+J¡ = (Jx + iJy) (Jx ¡ iJy) = J2x + J2y ¡ i [Jx; Jy]= J2x + J
2y + Jz = J
2 ¡ Jz(Jz ¡ 1) (5.89)
176 Angular Momentum and Rotations
and
J¡J+ = (Jx ¡ iJy) (Jx + iJy) = J2x + J2y + i [Jx; Jy]= J2x + J
2y ¡ Jz = J2 ¡ Jz(Jz + 1): (5.90)
These two results imply the relation
J2 =1
2[J+J¡ + J¡J+] + J2z : (5.91)
With these relations we now deduce allowed values in the spectrum of J2 andJz. We assume, at …rst, the existence of at least one nonzero eigenvector jj;mi of J2 andJz with angular momentum (j;m); where, consistent with our previous discussion, theeigenvalues of J2 satisfy the inequalities
j(j + 1) ¸ 0 j ¸ 0: (5.92)
Using this, and the commutation relations, we now prove that the eigenvalue m must lie,for a given value of j; in the range
j ¸m ¸ ¡j: (5.93)
To show this, we consider the vectors J+jj;mi and J¡jj;mi; whose squared norms arejjJ+jj;mijj2 = hj;mjJ¡J+jj;mi (5.94)
jjJ¡jj;mijj2 = hj;mjJ+J¡jj;mi: (5.95)
Using (5.89) and (5.90) these last two equations can be written in the form
jjJ+jj;mijj2 = hj;mjJ2 ¡ Jz(Jz + 1)jj;mi = [j(j + 1)¡m(m+ 1)] hj;mjj;mi (5.96)
jjJ¡jj;mijj2 = hj;mjJ2 ¡ Jz(Jz ¡ 1)jj;mi = [j(j + 1)¡m(m¡ 1)] hj;mjj;mi (5.97)
Now, adding and subtracting a factor of jm from each parenthetical term on the right,these last expressions can be factored into
jjJ+jj;mijj2 = (j ¡m)(j +m+ 1)jjjj;mijj2 (5.98)
jjJ¡jj;mijj2 = (j +m)(j ¡m+ 1)jjjj;mijj2: (5.99)
For these quantities to remain positive de…nite, we must have
(j ¡m)(j +m+ 1) ¸ 0 (5.100)
and(j +m)(j ¡m+ 1) ¸ 0: (5.101)
For positive j; the …rst inequality requires that
j ¸m and m ¸ ¡(j + 1) (5.102)
and the second thatm ¸ ¡j and j + 1 ¸ m: (5.103)
All four inequalities are satis…ed if and only if
j ¸m ¸ ¡j; (5.104)
Eigenstates and Eigenvalues of Angular Momentum Operators 177
which shows, as required, that the eigenvalue m of Jz must lie between §j.Having narrowed the range for the eigenvalues of Jz; we now show that the vector
J+jj;mi vanishes if and only if m = j; and that otherwise J+jj;mi is an eigenvector of J2and Jz with angular momentum (j;m+ 1); i.e., it is associated with the same eigenvaluej(j + 1) of J2; but is associated with an eigenvalue of Jz increased by one, as a result ofthe action of J+.
To show the …rst half of the statement, we note from (5.98), that
jjJ+jj;mijj = (j ¡m)(j +m+ 1)jjjj;mijj2: (5.105)
Given the bounds onm; it follows that J+jj;mi vanishes if and only ifm = j: To prove thesecond part, we use the commutation relation [Jz; J+] = J+ to write JzJ+ = J+Jz + J+and thus
JzJ+jj;mi = (J+Jz + J+)jj;mi = (m+ 1)J+jj;mi; (5.106)
showing that J+jj;mi is an eigenvector of Jz with eigenvalue m + 1: Also, because[J2; J+] = 0;
J2J+jj;mi = J+J2jj;mi = j(j + 1)J+jj;mi (5.107)
showing that if m 6= j; then the vector J+jj;mi is an eigenvector of J2 with eigenvaluej(j + 1):
In a similar fashion, using (5.99) we …nd that
jjJ¡jj;mijj = (j +m)(j ¡m+ 1)jjjj;mijj2: (5.108)
Given the bounds onm; this proves that the vector J¡jj;mi vanishes if and only ifm = ¡j;while the commutation relations [Jz; J¡] = ¡J¡ and [J2; J¡] = 0 imply that
JzJ¡jj;mi = (J¡Jz ¡ Jz)jj;mi = (m¡ 1)J¡jj;mi (5.109)
J2J¡jj;mi = J¡J2jj;mi = j(j + 1)J¡jj;mi: (5.110)
Thus, when m 6= ¡j; the vector J¡jj;mi is an eigenvector of J2 and Jz with angularmomentum (j;m¡ 1):
Thus, J+ is referred to as the raising operator, since it acts to increase the com-ponent of angular momentum along the z-axis by one unit and J¡ is referred to as thelowering operator. Neither operator has any e¤ect on the total angular momentum of thesystem, as represented by the quantum number j labeling the eigenvalues of J2:
We now proceed to restrict even further the spectra of J2 and Jz: We note, e.g.,from our preceding analysis that, given any vector jj;mi of angular momentum (j;m) wecan produce a sequence
J+jj;mi; J2+jj;mi; J3+jj;mi; ¢ ¢ ¢ (5.111)
of mutually orthogonal eigenvectors of J2 with eigenvalue j(j + 1) and of Jz with eigen-values
m; (m+ 1) ; (m+ 2) ; ¢ ¢ ¢ : (5.112)
This sequence must terminate, or else produce eigenvectors of Jz with eigenvalues violatingthe upper bound in Eq. (5.93). Termination occurs when J+ acts on the last nonzerovector of the sequence, Jn+jj;mi say, and takes it on to the null vector. But, as we haveshown, this can only occur if m + n = j; i.e., if Jn+jj;mi is an eigenvector of angularmomentum (j;m+ n) = (j; j): Thus, there must exist an integer n such that
n = j ¡m: (5.113)
178 Angular Momentum and Rotations
This, by itself, does not require that m or j be integers, only that the values of m changeby an integer amount, so that the di¤erence between m and j be an integer. But similararguments can be made for the sequence
J¡jj;mi; J2¡jj;mi; J3¡jj;mi; ¢ ¢ ¢ (5.114)
which will be a series of mutually orthogonal eigenvectors of J2 with eigenvalue j(j + 1)and of Jz with eigenvalues
m; (m¡ 1); (m¡ 2); ¢ ¢ ¢ : (5.115)
Again, termination is required to now avoid producing eigenvectors of Jz with eigenvaluesviolating the lower bound in Eq. (5.93). Thus, the action of J¡ on the last nonzero vectorof the sequence, Jn
0+ jj;mi say, is to take it onto the null vector. But this only occurs if
m¡n0 = ¡j; i.e., if Jn0+ jj;mi is an eigenvector of angular momentum (j;m¡n0) = (j;¡j):Thus there exists an integer n0 such that
n0 = j +m: (5.116)
Adding these two relations, we deduce that there exists an integer N = n+ n0 such that2j = n+ n0 = N , or
j =N
2: (5.117)
Thus, j must be either an integer or a half-integer. If N is an even integer, thenj is itself an integer and must be contained in the set j 2 f0; 1; 2; ¢ ¢ ¢ g: For this situation,the results of the proceeding analysis indicate that m must also be an integer and, fora given integer value of j; the values m must take on each of the 2j + 1 integer valuesm = 0;§1;§2; ¢ ¢ ¢§ j: In this case, j is said to be an integral value of angular momentum.
If N is an odd integer, then j di¤ers from an integer by 1=2; i.e., it is contained inthe set j 2 f1=2; 3=2; 5=2; ¢ ¢ ¢ g; and is said to be half-integral (short for half-odd-integral).For a given half-integral value of j; the values of m must then take on each of the 2j + 1half-odd-integer values m = §1=2; ¢ ¢ ¢ ;§j.
Thus, we have deduced the values of j and m that are consistent with the com-mutation relations (5.79). In particular, the allowed values of j that can occur are thenon-negative integers and the positive half-odd-integers. For each value of j; there arealways (at least) 2j + 1 fold mutually-orthogonal eigenvectors
fjj;mi j m = ¡j;¡j + 1; ¢ ¢ ¢ ; jg (5.118)
of J2 and Jz corresponding to the same eigenvalue j(j+1) of J2, but di¤erent eigenvaluesm of Jz (the orthogonality of the di¤erent vectors in the set follows from the fact that theyare eigenvectors of the Hermitian observable Jz corresponding to di¤erent eigenvalues.)
In any given problem involving an angular momentum ~J it must be determinedwhich of the allowed values of j and howmany subspaces for each such value actually occur.All of the integer values of angular momentum do, in fact, arise in the study of the orbitalangular momentum of a single particle, or of a group of particles. Half-integral values ofangular momentum, on the other hand, are invariably found to have their source in thehalf-integral angular momenta associated with the internal or spin degrees of freedom ofparticles that are anti-symmetric under exchange, i.e., fermions. Bosons, by contrast, areempirically found to have integer spins. This apparently universal relationship betweenthe exchange symmetry of identical particles and their spin degrees of freedom has actuallybeen derived under a rather broad set of assumptions using the techniques of quantum…eld theory.
Orthonormalization of Angular Momentum Eigenstates 179
The total angular momentum of a system of particles will generally have con-tributions from both orbital and spin angular momenta and can be either integral orhalf-integral, depending upon the number and type of particles in the system. Thus, gen-erally speaking, there do exist di¤erent systems in which the possible values of j and mdeduced above are actually realized. In other words, there appear to be no superselectionrules in nature that further restrict the allowed values of angular momentum from thoseallowed by the fundamental commutation relations.
5.7 Orthonormalization of Angular Momentum Eigenstates
We have seen in the last section that, given any eigenvector jj;mi with angular momen-tum (j;m); it is possible to construct a set of 2j + 1 common eigenvectors of J2 and Jzcorresponding to the same value of j, but di¤erent values of m: Speci…cally, the vectorsobtained by repeated application of J+ to jj;mi will produce a set of eigenvectors of Jzwith eigenvaluesm+1;m+2; ¢ ¢ ¢ ; j; while repeated application of J¡ to jj;mi will producethe remaining eigenvectors of Jz with eigenvalues m ¡ 1;m ¡ 2; ; ¢ ¢ ¢ ¡ j: Unfortunately,even when the original angular momentum eigenstate jj;mi is suitably normalized, theeigenvectors vectors obtained by application of the raising and lowering operators to thisstate are not. In this section, therefore, we consider the construction of a basis of nor-malized angular momentum eigenstates. To this end, we restrict our use of the notationjj;mi so that it refers only to normalized states. We can then express the action of J+on such a normalized state in the form
J+jj;mi = ¸mjj;m+ 1i (5.119)
where ¸m is a constant, and jj;m+1i represents, according to our de…nition, a normalizedstate with angular momentum (j;m+1). We can determine the constant ¸m by consideringthe quantity
jjJ+jj;mijj2 = hj;mjJ¡J+jj;mi = j¸mj2 (5.120)
which from the analysis following Eqs. (5.98) and (5.99), and the assumed normalization ofthe state jj;mi; reduces to j¸mj2 = j(j+1)¡m(m+1): Choosing ¸m real and positive,thisimplies the following relation
J+jj;mi =pj(j + 1)¡m(m+ 1) jj;m+ 1i (5.121)
between normalized eigenvectors of J2 and Jz di¤ering by one unit of angular momentumalong the z axis. A similar analysis applied to the operator J¡ leads to the relation
J¡jj;mi =pj(j + 1)¡m(m¡ 1) jj;m¡ 1i: (5.122)
These last two relations can also be written in the sometimes more convenient form
jj;m+ 1i = J+jj;mipj(j + 1)¡m(m+ 1) (5.123)
jj;m¡ 1i = J¡jj;mipj(j + 1)¡m(m¡ 1) : (5.124)
or
jj;m§ 1i = J§jj;mipj(j + 1)¡m(m§ 1) : (5.125)
Now, if J2 and Jz do not comprise a complete set of commuting observables for thespace on which they are de…ned, then other commuting observables (e.g., the energy) will
180 Angular Momentum and Rotations
be required to form a uniquely labeled basis of orthogonal eigenstates, i.e., to distinguishbetween di¤erent eigenvectors of J2 and Jz having the same angular momentum (j;m): Ifwe let ¿ denote the collection of quantum numbers (i.e., eigenvalues) associated with theother observables needed along with J2 and Jz to form a complete set of observables, thenthe normalized basis vectors of such a representation can be written in the form fj¿ ; j;mig :Typically, many representations of this type are possible, since we can always form linearcombinations of vectors with the same values of j and m to generate a new basis set.Thus, in general, basis vectors having the same values of ¿ and j; but di¤erent valuesof m need not obey the relationships derived above involving the raising and loweringoperators. However, as we show below, it is always possible to construct a so-calledstandard representation, in which the relationships (5.123) and (5.124) are maintained.
To construct such a standard representation, it su¢ces to work within each eigen-subspace S(j) of J2 with …xed j; since states with di¤erent values of j are automaticallyorthogonal (since J2 is Hermitian). Within any such subspace S(j) of J2, there are alwayscontained even smaller eigenspaces S(j;m) spanned by the vectors fj¿; j;mig of …xed jand …xed m: We focus in particular on the subspace S(j; j) containing eigenvectors ofJz for which m takes its highest value m = j, and denote by fj¿ ; j; jig a complete setof normalized basis vectors for this subspace, with the index ¿ distinguishing betweendi¤erent orthogonal basis vectors with angular momentum (j; j). By assumption, then,for the states in this set,
h¿ ; j; jj¿ 0; j; ji = ±¿;¿ 0 (5.126)
For each member j¿ ; j; ji of this set, we now construct the natural sequence of 2j+1 basisvectors by repeated application of J¡; i.e., using (5.124) we set
j¿ ; j; j ¡ 1i = J¡j¿; j; jipj(j + 1)¡ j(j ¡ 1) =
J¡j¿ ; j; jip2j
(5.127)
and the remaining members of the set according to the relation
j¿ ; j;m¡ 1i = J¡j¿; j;mipj(j + 1)¡m(m¡ 1) ; (5.128)
terminating the sequence with the vector j¿ ; j;¡ji: Since the members j¿ ; j;mi of this set(with ¿ and j …xed and m = ¡j; ¢ ¢ ¢ ; j) are eigenvectors of Jz corresponding to di¤erenteigenvalues, they are mutually orthogonal and, by construction, properly normalized.It is also straightforward to show that the vectors j¿ ; j;mi generated in this way fromthe basis vector j¿ ; j; ji of S(j; j) are orthogonal to the vectors j¿ 0; j;mi generated froma di¤erent basis vector j¿ 0; j; ji of S(j; j). To see this, we consider the inner producth¿ ; j;m¡ 1j¿ 0; j;m¡ 1i and, using the adjoint of (5.128),
h¿; j;m¡ 1j = h¿ ; j;mjJ+pj(j + 1)¡m(m¡ 1) ; (5.129)
we …nd that
h¿ ; j;m¡ 1j¿ 0; j;m¡ 1i = h¿ ; j;mjJ+J¡j¿ 0; j;mij(j + 1)¡m(m¡ 1) = h¿; j;mj¿
0; j;mi (5.130)
where we have used (5.99) to evaluate the matrix element of J+J¡: This shows that,if j¿ 0; j;mi and j¿ ; j;mi are orthogonal, then so are the states generated from them byapplication of J¡: Since the basis states j¿; j; ji and j¿ 0; j; ji used to start each sequenceare orthogonal, by construction, so, it follows, are any two sequences of basis vectors
Orthonormalization of Angular Momentum Eigenstates 181
so produced, and so also are the subspaces S(¿ ; j) and S(¿ 0; j) spanned by those basisvectors. Proceeding in this way for all values of j a standard representation of basisvectors for the entire space is produced. Indeed, the entire space can be written as adirect sum of the subspaces S(¿; j) so formed for each j; i.e., we can write
S = S(j)© S(j0)© S(j00) + ¢ ¢ ¢with a similar decomposition
S(j) = S(¿ ; j)© S(¿ 0; j)© S(¿ 00; j) + ¢ ¢ ¢for the eigenspaces S(j) of j2. The basis vectors for this representation satisfy the obviousorthonormality and completeness relations
h¿ 0; j0;m0j¿ ; j;mi = ±¿ 0;¿±j0;j±m0;m (5.131)X¿;j;m
j¿ ; j;mih¿ ; j;mj = 1: (5.132)
The matrices representing the components of the angular momentum operators are easilycomputed in any standard representation. In particular, it is easily veri…ed that the matrixelements of J2 are given in any standard representation by the expression
h¿ 0; j0;m0jJ2j¿; j;mi = j(j + 1)±¿ 0;¿±j0;j±m0;m (5.133)
while the components of ~J have the following matrix elements
h¿ 0; j0;m0jJzj¿ ; j;mi = m±¿ 0;¿±j0;j±m0;m (5.134)
h¿ 0; j0;m0jJ§j¿ ; j;mi =pj(j + 1)¡m(m§ 1)±¿ 0;¿±j0;j±m0;m§1: (5.135)
The matrices representing the Cartesian components Jx and Jy can then be constructedfrom the matrices for J+ and J¡; using relations (5.84), i.e.
h¿ 0; j0;m0jJxj¿ ; j;mi =1
2±¿ 0;¿±j0;j
hpj(j + 1)¡m(m¡ 1)±m0;m¡1
+pj(j + 1)¡m(m+ 1)±m0;m+1
i(5.136)
h¿ 0; j0;m0jJyj¿ ; j;mi =i
2±¿ 0;¿±j0;j
hpj(j + 1)¡m(m¡ 1)±m0;m¡1
¡pj(j + 1)¡m(m+ 1)±m0;m+1
i(5.137)
Clearly, the simplest possible subspace of …xed total angular momentum j is onecorresponding to j = 0; which according to the results derived above must be of dimension2j+1 = 1: Thus, the one basis vector j¿ ; 0; 0i in such a space is a simultaneous eigenvectorof J2 and of Jz with eigenvalue j(j+1) =m = 0. Such a state, as it turns out, is also aneigenvector of Jx and Jy (indeed of ~J itself). The 1£ 1 matrices representing J2; Jz; J§;Jx; and J¡ in such a space are all identical to the null operator.
The next largest possible angular momentum subspace corresponds to the valuej = 1=2; which coincides with the 2j+1 = 2 dimensional spin space of electrons, protons,and neutrons, i.e., particles of spin 1/2. In general, the full Hilbert space of a singleparticle of spin s can be considered the direct product
S = Sspatial Sspin (5.138)
182 Angular Momentum and Rotations
of the Hilbert space describing the particle’s motion through space (a space spanned,e.g., by the position states j~ri) and a …nite dimensional space describing its internal spindegrees of freedom. The spin space Ss of a particle of spin s is by de…nition a 2s + 1dimensional space having as its fundamental observables the components Sx; Sy; and Szof a spin angular momentum vector ~S: In keeping with the analysis of Sec. (5.3), theseoperators characterize the way that the spin part of the state vector transforms underrotations and, as such, satisfy the standard angular momentum commutation relations,i.e.,
[Si; Sj ] = iXk
"ijkSk: (5.139)
Thus, e.g., an ONB for the space of one such particle consists of the states j~r;msi; whichare eigenstates of the position operator ~R; the total square of the spin angular momentumS2; and the component of spin Sz along the z-axis according to the relations
~Rj~r;msi = ~rj~r;msi (5.140)
S2j~r;msi = s(s+ 1)j~r;msi (5.141)
Szj~r;msi =msj~r;msi: (5.142)
(As is customary, in these last expressions the label s indicating the eigenvalue of S2 hasbeen suppressed, since for a given class of particle s does not change.) The state vectorjÃi of such a particle, when expanded in such a basis, takes the form
jÃi =sX
ms=¡s
Zd3r j~r;msih~r;msjÃi =
sXms=¡s
Zd3r Ãms
(~r)j~r;msi (5.143)
and thus has a “wave function” with 2s + 1 components Ãms(~r). As one would expect
from the de…nition of the direct product, operators from the spatial part of the space haveno e¤ect on the spin part and vice versa. Thus, spin and spatial operators automaticallycommute with each other. In problems dealing only with the spin degrees of freedom,therefore, it is often convenient to simply ignore the part of the space associated with thespatial degrees of freedom (as the spin degrees of freedom are often generally ignored inexploring the basic features of quantum mechanics in real space).
Thus, the spin space of a particle of spin s = 1=2, is spanned by two basis vectors,often designated j+i and j¡i; with
S2j§i = 1
2
µ1
2+ 1
¶j§i = 3
4j§i (5.144)
and
Szj§i = §12j§i: (5.145)
The matrices representing the di¤erent components of ~S within a standard representationfor such a space are readily computed from (5.133)-(5.137),
S2 =3
4
µ1 00 1
¶(5.146)
Sx =1
2
µ0 11 0
¶Sy =
1
2
µ0 ¡ii 0
¶Sz =
1
2
µ1 00 ¡1
¶(5.147)
Orbital Angular Momentum Revisited 183
S+ =
µ0 10 0
¶S¡ =
µ0 01 0
¶(5.148)
The Cartesian components Sx; Sy; and Sz are often expressed in terms of the so-calledPauli ¾ matrices
¾x =
µ0 11 0
¶¾y =
µ0 ¡ii 0
¶¾z =
µ1 00 ¡1
¶(5.149)
in terms of which Si = 12¾i. The ¾ matrices have a number of interesting properties that
follow from their relation to the angular momentum operators.
5.8 Orbital Angular Momentum Revisited
As an additional example of the occurrance of angular momentum subspaces with integralvalues of j we consider again the orbital angular momentum of a single particle as de…nedby the operator ~= ~R£ ~K with Cartesian components `i =
Pj;k "ijkXjKk: In the position
representation these take the form of di¤erential operators
`i = ¡iXj;k
"ijkxj@
@xk: (5.150)
As it turns out, in standard spherical coordinates (r; µ; Á); where
x = r sin µ cosÁ y = r sin µ sinÁ z = r cos µ; (5.151)
the components of ~ take a form which is independent of the radial variable r. Indeed,using the chain rule it is readily found that
`x = i
µsinÁ
@
@µ+ cosÁ cot µ
@
@Á
¶(5.152)
`y = i
µ¡ cosÁ @
@µ+ sinÁ cot µ
@
@Á
¶(5.153)
`z = ¡i @@Á: (5.154)
In keeping with our previous development, it is useful to construct from `x and `y theraising and lowering operators
`§ = `x § i`y (5.155)
which (5.152) and (5.153) reduce to
`§ = e§iÁ·§ @
@µ+ i cot µ
@
@Á
¸: (5.156)
From these it is also straightforward to construct the di¤erential operator
`2 = ¡·@2
@µ2+ cot µ
@
@µ+
1
sin2 µ
@2
@Á2
¸(5.157)
representing the total square of the angular momentum.Now we are interested in …nding common eigenstates of `2 and `z: Since these
operators are independent of r; it su¢ces to consider only the angular dependence. It
184 Angular Momentum and Rotations
is clear, in other words, that the eigenfunctions of these operators can be written in theform
Ãl;m(r; µ; Á) = f(r)Yml (µ; Á) (5.158)
where f(r) is any acceptable function of r and the functions Yml (µ; Á) are solutions tothe eigenvalue equations
`2Yml (µ; Á) = l(l+ 1)Y ml (µ; Á)
`zYml (µ; Á) = mYml (µ; Á) : (5.159)
Thus, it su¢ces to …nd the functions Yml (µ; Á) ; which are, of course, just the sphericalharmonics.
More formally, we can consider the position eigenstates j~ri as de…ning directproduct states
j~ri = jr; µ; Ái = jri jµ; Ái; (5.160)
i.e., the space can be decomposed into a direct product of a part describing the radialpart of the wave function and a part describing the angular dependence. The angularpart represents the space of functions on the unit sphere, and is spanned by the “angularposition eigenstates” jµ; Ái: In this space, an arbitrary function Â(µ; Á) on the unit sphereis associated with a ket
jÂi =Zd Â(µ; Á)jµ; Ái =
Z 2¼
0
dÁ
Z ¼
0
dµ sin µ Â(µ; Á)jµ; Ái (5.161)
where Â(µ; Á) = hµ; ÁjÂi and the integration is over all solid angle, d = sin µ dµ dÁ: Thestates jµ; Ái form a complete set of states for this space, and soZ
d jµ; Áihµ; Áj = 1: (5.162)
The normalization of these states is slightly di¤erent than the usual Dirac normaliza-tion, however, because of the factor associated with the transformation from Cartesian tospherical coordinates. To determine the appropriate normalization we note that
Â(µ0; Á0) = hµ0; Á0jÂi =Z 2¼
0
dÁ
Z ¼
0
dµ sin µ Â(µ; Á)hµ0; Á0jµ; Ái (5.163)
which leads to the identi…cation ±(Á¡ Á0)±(µ ¡ µ0) = sin µ hµ0; Á0jµ; Ái; or
hµ0; Á0jµ; Ái = 1
sin µ±(Á¡ Á0)±(µ ¡ µ0) = ±(Á¡ Á0)±(cos µ ¡ cos µ0): (5.164)
Clearly, the components of ~ are operators de…ned on this space and so we denote by jl;mithe appropriate eigenstates of the Hermitian operators `2 and `z within this space (thisassumes that l;m are su¢cient to specify each eigenstate, which, of course, turns out tobe true). By assumption, then, these states satisfy the eigenvalue equations
`2jl;mi = l(l+ 1)jl;mi`zjl;mi = mjl;mi (5.165)
and can be expanded in the angular “position representation”, i.e.,
jl;mi =Z± jµ; Áihµ; Ájl;mi =
Z± Y ml (µ; Á) jµ; Ái (5.166)
Orbital Angular Momentum Revisited 185
where the functionsY ml (µ; Á) = hµ; Ájl;mi (5.167)
are clearly the same as those introduced above, and will turn out to be the sphericalharmonics.
To obtain the states jl;mi (or equivalently the functions Yml ) we proceed in threestages. First, we determine the general Á-dependence of the solution from the eigenvalueequation for `z: Then, rather than solving the second order equation for `2 directly, wedetermine the general form of the solution for the states jl; li having the largest componentof angular momentum along the z-axis consistent with a given value of l. Finally, weuse the lowering operator `¡ to develop a general prescription for constructing arbitraryeigenstates of `2 and `z:
The Á-dependence of the eigenfunctions in the position representation follows fromthe simple form taken by the operator `z in this representation. Indeed, using (5.154), itfollows that
`zYml (µ; Á) = ¡i
@
@ÁY ml (µ; Á) = mY
ml (µ; Á); (5.168)
which has the general solution
Y ml (µ; Á) = Fml (µ)e
imÁ: (5.169)
Single-valuedness of the wave function in this representation imposes the requirement thatYml (µ; Á) = Y
ml (µ; Á+ 2¼); which leads to the restriction m 2 f0;§1;§2; ¢ ¢ ¢ g: Thus, for
the case of orbital angular momentum only integral values of m (and therefore l) canoccur. (We have yet to show that all integral values of l do, in fact, occur, however.)
From this result we now proceed to determine the eigenstates jl; li; as representedby the wave functions
Y ll (µ; Á) = Fll (µ)e
ilÁ: (5.170)
To this end, we recall the general result that any such state of maximal angular momentumalong the z axis is taken by the raising operator onto the null vector, i.e., `+jl; li = 0. Inthe position representation, using (5.156), this takes the form
hµ; Áj`+jl; li = eiÁ·@
@µ+ i cot µ
@
@Á
¸Y ll (µ; Á)
= eiÁ·@
@µ+ i cot µ
@
@Á
¸F ll (µ)e
ilÁ = 0: (5.171)
Performing the Á derivative reduces this to a …rst order equation for F ll (µ), i.e.,
dF ll (µ)
dµ= l cot µF ll (µ) (5.172)
dF llF ll
= ld (sin µ)
sin µ(5.173)
which integrates to give, up to an overall multiplicative constant, a single linearly inde-pendent solution
F ll (µ) = cl sinl µ (5.174)
for each allowed value of l. Thus, all values of l consistent with the integer values of mdeduced above give acceptable solutions. Up to normalization we have, therefore, for eachl = 0; 1; 2 ¢ ¢ ¢ ; the functions
Y ll (µ; Á) = cl sinl µ eilÁ: (5.175)
186 Angular Momentum and Rotations
The appropriate normalization for these functions follows from the relation
hl;mjl;mi = 1 (5.176)
which in the position representation becomesZd hl;mjµ; Áihµ; Ájl;mi =
Zd [Y ml (µ; Á)]
¤ Y m0
l0 (µ; Á)
=
Z 2¼
0
dÁ
Z ¼
0
dµ sin µ jY ml (µ; Á)j2 = 1 (5.177)
Substituting in the function Y ll (µ; Á) = cl sinl µ eilÁ; the magnitude of the constants cl can
be determined by iteration, with the result that
jclj =s(2l + 1)!!
4¼ (2l)!!=
1
2ll!
r(2l + 1)!
4¼; (5.178)
where the double factorial notation is de…ned on the postive integers as follows:
n!! =
8<: n(n¡ 2)(n¡ 4) ¢ ¢ ¢ (2) if n an even integern(n¡ 2)(n¡ 4) ¢ ¢ ¢ (1) if n an odd integer1 if n = 0
(5.179)
With the phase of cl; (chosen so that Y 0l (0; 0) is real and positive) given by the relationcl = (¡1)l jclj we have the …nal form for the spherical harmonic of order (l; l); i.e.
Y ll (µ; Á) =(¡1)l2ll!
r(2l + 1)!
4¼sinl µ eilÁ: (5.180)
From this, the remaining spherical harmonics of the same order l can be generatedthrough application of the lowering operator, e.g., through the relation
jl;m¡ 1i = `¡jl;mipl(l + 1)¡m(m¡ 1) (5.181)
which, in the position representation takes the form
Y m¡1l (µ; Á) = e¡iÁ·¡ @
@µ¡m cot µ
¸Yml (µ; Á) : (5.182)
In fact, it straightforward to derive the following expression
jl;mi =s
(l +m)!
(2l)!(l +m)![`¡]
l¡m jl; li (5.183)
relating an arbitrary state jl;mi to the state jl; li which we have explicitly found. Thislast relation is straightforward to prove by induction. We …rst assume that it holds forsome value of m and then consider
jl;m¡1i = `¡jl;mipl(l+ 1)¡m(m¡ 1) =
1pl(l + 1)¡m(m¡ 1)
s(l ¡m)!
(2l)!(l +m)![`¡]
l¡m+1 jl; li:(5.184)
Orbital Angular Momentum Revisited 187
But, as we have noted before, l(l+1)¡m(m¡1) = (l +m) (l ¡m+ 1) : Substituting thisinto the radical on the right and canceling the obvious terms which arise we …nd that
jl;m¡ 1i =s
[l ¡ (m¡ 1)]!(2l)!(l +m¡ 1)! [`¡]
l¡(m¡1) jl; li (5.185)
which is of the same form as the expression we are trying to prove withm!m¡1: Thus,if it is true for any m it is true for all values of m less than the original. We then note thatthe expression is trivially true for m = l; which completes the proof. Thus, the sphericalharmonic of order (l;m) can be expressed in terms of the one of order (l; l) in the form
Y ml (µ; Á) =
s[l ¡ (m¡ 1)]!(2l)!(l+m¡ 1)!e
¡i(l¡m)Á·¡ @
@µ+ i cot µ
@
@Á
¸l¡(m¡1)Y ll (µ; Á): (5.186)
It is not our intention to provide here a complete derivation of the properties ofthe spherical harmonics, but rather to show how they …t into the general scheme we havedeveloped regarding angular momentum eigenstates in general. To round things out a bitwe mention without proof a number of their useful properties.
1. Parity - The parity operator ¦ acts on the eigenstates of the position representationand inverts them through the origin, i.e., ¦j~ri = j¡~ri: It is straightforward to showthat in the position representation this takes the form ¦Ã(~r) = Ã(¡~r): In sphericalcoordinates it is also easily veri…ed that under the parity operation r! r; µ ! ¼¡µ;and Á! Á+ ¼: Thus, for functions on the unit sphere, ¦f(µ; Á) = f(¼ ¡ µ; Á+ ¼):The parity operator commutes with the components of ~ and with `2: Indeed, thestates jl;mi are eigenstates of parity and satisfy the eigenvalue equation
¦jl;mi = (¡1)l jl;mi (5.187)
which implies for the spherical harmonics that
¦Y ml (µ; Á) = Yml (¼ ¡ µ; Á+ ¼) = (¡1)l Y ml (µ; Á): (5.188)
2. Complex Conjugation - It is straightforward to show that
[Y ml (µ; Á)]¤ = (¡1)mY ¡ml (µ; Á): (5.189)
This allows spherical harmonics with m < 0 to be obtained very simply from thosewith m > 0.
3. Relation to Legendre Functions - The spherical harmonics with m = 0 aredirectly related to the Legendre polynomials
Pl(u) =(¡1)l2ll!
dl
dul¡1¡ u2¢l (5.190)
through the relation
Y 0l (µ; Á) =
r(2l + 1)!
4¼Pl(cos µ): (5.191)
The other spherical harmonics with m > 0 are related to the associated Legendrefunctions
Pml (u) =p(1¡ u2)m dmPl(u)
dum; (5.192)
through the relation
Y ml (µ; Á) = (¡1)ms(2l+ 1)!(l¡m)!4¼ (l +m)!
Pml (cos µ)eimÁ: (5.193)
188 Angular Momentum and Rotations
5.9 Rotational Invariance
As we have seen, the behavior exhibited by a quantum system under rotations can usuallybe connected to a property related to the angular momentum of the system. In thischapter we consider in some detail the consequences of rotational invariance, that is, weexplore just what is implied by the statement that a certain physical state or quantity isunchanged as a result of rotations imposed upon the system. To begin the discussion weintroduce the important idea of rotationally invariant subspaces.
5.9.1 Irreducible Invariant Subspaces
In our discussion of commuting or compatible observables we encountered the idea ofglobal invariance and saw, e.g., that the eigenspaces Sa of an observable A are globallyinvariant with respect to any operator B that commutes with A: It is useful to extendthis idea to apply to more than one operator at a time. We therefore introduce the ideaof invariant subspaces.
A subspace S0 of the state space S is said to be invariant with respect tothe action of a set G = fR1; R2; ¢ ¢ ¢ g of operators if, for every jÃi in S0; the vectorsR1jÃi; R2jÃi; ¢ ¢ ¢ are all in S0 as well. S0 is than said to be an invariant subspace of thespeci…ed set of operators. The basic idea here is that the operators Ri all respect theboundaries of the subspace S0; in the sense that they never take a state in S0 onto a stateoutside of S0.
With this de…nition, we consider a quantum mechanical system with state spaceS; characterized by total angular momentum ~J: It is always possible to express the statespace S as a direct sum of orthogonal eigenspaces associated with any observable. (Recallthat a space can be decomposed into a direct sum of two or more orthogonal subspacesif any vector in the space can be written as a linear combination of vectors from eachsubspace.) In the present context we consider the decomposition
S = S(j)© S(j0)© S(j00) + ¢ ¢ ¢ (5.194)
of our original space S into eigenspaces S(j) of the operator J2. Now each one of thespaces S(j) associated with a particular eigenvalue j(j+1) of J2 can, itself, be decomposedinto a direct sum
S(j) = S(¿ ; j)© S(¿ 0; j)© S(¿ 00; j) + ¢ ¢ ¢ (5.195)
of 2j +1 dimensional subspaces S(¿; j) associated with a standard representation for thespace S, i.e., the vectors in S(¿ ; j) comprise all linear combinations
jÃi =jX
m=¡jÃmj¿ ; j;mi (5.196)
of the 2j + 1 basis states j¿ ; j;mi with …xed ¿ and j. We now show that each of thesespaces S(¿; j) is invariant under the action of the operator components
nJu = ~J ¢ u
oof
the total angular momentum. This basically follows from the the way that such a standardrepresentation is constructed. We just need to show that the action of any componentof ~J on such a vector takes it onto another vector in the same space, i.e., onto a linearcombination of the same basis vectors. But the action of the operator Ju = ~J ¢u =Pi uiJiis completely determined by the action of the three Cartesian components of the vectoroperator ~J . Clearly, however, the vector
JzjÃi =jX
m=¡jmÃmj¿; j;mi (5.197)
Rotational Invariance 189
lies in the same subspace S(¿ ; j) as jÃi:Moreover, Jx and Jy are simple linear combinationsof J§; which take such a state onto
J§jÃi =jX
m=¡j
pj(j + 1)¡m(m§ 1)Ãmj¿ ; j;m§ 1i (5.198)
which is also in the subspace S(¿ ; j). Thus it is clear the the components of ~J cannottake an arbitrary state jÃi in S(¿; j) outside the subspace. Hence S(¿ ; j) is an invariantsubspace of the angular momentum operators. Indeed, it is precisely this invariance thatleads to the fact the the matrices representing the components of ~J are block-diagonal inany standard representation.
It is not di¢cult to see that this invariance with respect to the action of theoperator Ju extends to any operator function F (Ju). In particular, it extends to theunitary rotation operators Uu(®) = exp (¡i®Ju) which are simple exponential functionsof the components of ~J . Thus, we conclude that the spaces S(¿ ; j) are also invariantsubspaces of the group of operators fUu(®)g : We say that each subspace S(¿ ; j) is aninvariant subspace of the rotation group, or that S(¿ ; j) is rotationally invariant. We notethat the eigenspace S(j) is also an an invariant subspace of the rotation group, since anyvector in it is a linear combination of vectors from the invariant subspaces S(¿ ; j) and sothe action of Uu(®) on an arbitrary vector in Sj is to take it onto another vector in S(j);with the di¤erent parts in each subspace S(¿ ; j) staying within that respective subspace.It is clear, however, that although S(j) is invariant with respect to the operators of therotation group, it can often be reduced or decomposed into lower dimensional invariantparts S(¿ ; j). It is is reasonable in light of this decomposability exhibited by S(j); to askwhether the rotationally invariant subspaces S(¿ ; j) of which S(j) is formed are similarlydecomposable. To answer this question we are led to the idea of irreducibility.
An invariant subspace S0 of a group of operators G = fR1; R2; ¢ ¢ ¢ g is said to beirreducible with respect to G (or is an irreducible invariant subspace of G) if, for everynon-zero vector jÃi in S0, the vectors fRijÃig span S0. Conversely, S0 is reducible ifthere exists a nonzero vector jÃi in S0 for which the vectors fRijÃig fail to span the space.Clearly, in the latter case the vectors spanned by the set fRijÃig form a subspace of S0
that is itself invariant with respect to G:We now answer the question we posed above, and show explicitly that any invari-
ant subspace S(¿ ; j) associated with a standard representation for the state space S is, infact, an irreducible invariant subspace of the rotation group fUu(®)g, i.e., S(¿; j) cannotbe decomposed into smaller invariant subspaces. To prove this requires several steps. Tobegin, we let
jÃi =jX
m=¡jÃmj¿ ; j;mi (5.199)
again be an arbitrary (nonzero) vector in S(¿; j) and we formally denote by SR thesubspace of S(¿ ; j) spanned by the vectors fUu(®)jÃig ; that is, SR is the subspace of allvectors that can be written as a linear combination of vectors obtained through a rotationof the state jÃi: It is clear that SR is contained in S(¿; j); since the latter is invariantunder rotations; the vectors fUu(®)jÃig must all lie inside S(¿ ; j). We wish to show thatin fact SR = S(¿; j): To do this we show that SR contains a basis for S(¿; j) and hencethe two spaces are equivalent.
To this end we note that the vectors JujÃi are all contained in SR: This followsfrom the form of in…nitesimal rotations Uu(±®) = 1¡ i±®Ju; which imply that
Ju =1
i®[1¡ Uu(±a)] = 1
i®[Uu(0)¡ Uu(±a)] : (5.200)
190 Angular Momentum and Rotations
Thus,
JujÃi = 1
i®[Uu(0)¡ Uu(±a)] jÃi = 1
i®[Uu(0)jÃi ¡ Uu(±a)jÃi] (5.201)
which is, indeed, a linear combination of the vectors fUu(®)jÃig, and thus is in the spaceSR spanned by such vectors. By a straightforward extension, it follows, that the vectors
J§jÃi = (Jx § iJy)jÃi=
1
i®[Ux(0)¡ Ux(±a)§ iUy(0)¨ iUy(±a)] jÃi (5.202)
are also in SR: In fact, any vector of the form
(J¡)q (J+)
P jÃi (5.203)
will be expressable as a linear combination of various products of Ux(0); Uy(0); Ux(±®);and Uy(±®) acting on jÃi. Since the product of any two such rotation operators is itselfa rotation, the result will be a linear combination of the vectors fUu(®)jÃig ; and so willalso lie in SR:
We now note, that since jÃi is a linear combination of the vectors j¿ ; j;mi, eachof which is raised or annihilated by the operator J+; there exists an integer P < 2j forwhich (J+)
P jÃi = ¸j¿ ; j; ji (in other words, we keep raising the components of jÃi up andannhilating them till the component that initially had the smallest value of Jz is the onlyone left.) It follows, therefore, that the basis state j¿ ; j; ji lies in the subspace SR. Weare now home free, since we can now repeatedly apply the lowering operator, remainingwithin SR with each application, to deduce that all of the basis vectors j¿; j;mi lie in thesubspace SR: Thus SR is a subspace of S(¿ ; j) that contains a basis for S(¿ ; j): The onlyway this can happen is if SR = S(¿ ; j):
It follows that the under the unitary transformation associated with a generalrotation, the basis vectors fj¿ ; j;mig of S(¿ ; j) are transformed into new basis vectorsfor the same invariant subspace. Indeed, it is not hard to see, based upon our earlierdescription of the rotation process, that a rotation R that takes the unit vector z ontoa new direction z0 will take the basis kets j¿ ; j;mzi, which are eigenstates of J2 andJz onto a new set of basis kets j¿ ; j;mz0i for the same space that are now eigenstatesof J2 and the component Jz0 of ~J along the new direction. These new vectors can,of course, be expressed as linear combinations of the original ones. The picture thatemerges is that, under rotations, the vectors j¿; j;mi transform (irreducibly) into linearcombinations of themselves. This transformation is essentially geometric in nature and isanalogous to the way that normal basis vectors in R3 transform into linear combinations ofone another. Indeed, by analogy, the coe¢cients of this linear transformation are identicalin any subspace of a standard representation having the same value of j; since the basisvectors of such a representation have been constructed using the angular momentumoperators in precisely the same fashion. This leads to the concept of rotation matrices,i.e., a set of standard matrices representing the rotation operators Uu(®) in terms of theire¤ect on the vectors within any irreducible invariant subspace S(¿; j). Just as with thematrices representing the components of ~J within any irreducible subspace S(¿ ; j), theelements of the rotation matrices will depend upon j and m but are indepndent of ¿ .Thus, e.g., a rotation UR of a basis ket j¿ ; j;mi of S(¿ ; j) results in a linear combination
Rotational Invariance 191
of such states of the form
URj¿ ; j;mi =
jXm0=¡j
j¿; j;m0ih¿ ; j;m0jURj¿; j;mi
=
jXm0=¡j
j¿; j;m0iR(j)m0;m (5.204)
where, as the notation suggests, the elements
R(j)m0;m = h¿; j;m0jURj¿ ; j;mi = hj;m0jURj; j;mi (5.205)
of the 2j + 1 dimensional rotation matrix are independent of ¿ : In terms of the elementsR(j)m0;m of the rotation matrices, the invariance of the subspaces S(¿ ; j) under rotationsleads to the general relation
h¿ 0; j0;m0jURj¿ ; j;mi = ±j0;j±¿ 0;¿R(j)m0;m: (5.206)
These matrices are straightforward to compute for low dimensional subspaces,and general formulas have been developed for calculating the matrices for rotations asso-ciated with the Euler angles (®; ¯; °). For rotations about the z axis the matrices take aparticulalrly simple form, since the rotation operator in this case is a simple function ofthe operator Jz of which the states jj;mi are eigenstates. Thus, e.g.,
R(j)m0m(z; ®) = hj;m0je¡i®Jz jj;mi = e¡im®±m;m0 (5.207)
In a subspace with j = 1=2; for example, this takes the form
R(1=2)(z; ®) =
µe¡i®=2 00 e+i®=2
¶; (5.208)
while in a space with j = 1 we have
R(1)(z;®) =
0@ e¡i® 0 00 1 00 0 ei®
1A :The point is that, once the rotation matrices have been worked out for a given
value of j, they can be used for a standard representation of any quantum mechanicalsystem. Thus, e.g., we can deduce a transformation law associated with rotations of thespherical harmonics, i.e.,
R [Yml (µ; Á)] = hµ; ÁjURjl;mi
=lX
m0=¡lYm
0l (µ; Á)R
(l)m0;m: (5.209)
For rotations about the z-axis, this takes the form
Rz(®) [Yml (µ; Á)] = hµ; Áje¡i®`z jl;mi
= e¡im®Y ml (µ; Á) = e¡im®Fml (µ)e
imÁ
= Fml (µ)eim(Á¡®)
= Y ml (µ; Á¡ ®) (5.210)
which is readily con…rmed from simple geometric arguments.
192 Angular Momentum and Rotations
5.9.2 Rotational Invariance of States
We now consider a physical state of the system that is invariant under rotations, i.e., thathas the property that
URjÃi = jÃi (5.211)
for all rotations UR. This can be expressed in terms of the angular momentum of thesystem by noting that invariance under in…nitesimal rotations
Uu(±®)jÃi = jÃi ¡ i±®JujÃi = jÃi;
requires thatJujÃi = 0 (5.212)
for all u, which also implies that J2jÃi = 0: Thus, a state jÃi is rotationally invariant ifand only if it has zero angular momentum.
5.9.3 Rotational Invariance of Operators
If an observable Q is rotationally invariant it is, by our earlier de…nition, a scalar withrespect to rotations, and we can deduce the following:
1. [UR; Q] = 0
2. [Ju; Q] = 0 = [J2;Q]
3. There exists an ONB of eigenstates states fj¿; q; j;mig common to Jz; J2; and Q.4. The eigenvalues of q of Q within any irreducible subspace S(j) are (at least) 2j +1fold degenerate.
This degeneracy, referred to as a rotational or essential degeneracy, is straight-foward to show. Suppose that jqi is an eigenstate of Q; so that Qjqi = qjqi: Then
QURjqi = URQjqi = qURjqi: (5.213)
This shows that URjqi is an eigenstate of Q with the same eigenvalue. Of course not allthe states fURjqig are linearly independent. From this set of states we can form linearcombinations which are also eigenstates of J2 and Jz; and which can be partitioned intoirreducible invariant subspaces of well de…ned j: The 2j + 1 linearly independent basisstates associated with each such irreducible subspace S(q; j) are then 2j+1 fold degenerateeigenstates of Q.
As an important special case, suppose that the Hamiltonian of a quantum systemis a scalar with respect to rotations. We can then conclude that
1. H is rotationally invariant.
2. [UR;H] = [Ju;H] = [J2;H] = 0:
3. The components of ~J are constants of the motion, since
d
dthJui = i
~h[H;Ju]i = 0: (5.214)
4. The equations of motion are invariant under rotations. Thus, if jÃ(t)i is a solutionto µ
i~ ddt¡H
¶jÃ(t)i = 0 (5.215)
Addition of Angular Momenta 193
then
UR
µi~ ddt¡H
¶jÃ(t)i =
µi~ ddt¡H
¶URjÃ(t)i = 0 (5.216)
which shows that the rotated state URjÃ(t)i is also a solution to the equations ofmotion.
5. There exists an ONB of eigenstates states fjE; ¿; j;mig common to Jz; J2; and H.6. The eigenvalues E of H within any irreducible subspace are (at least) 2j + 1 folddegenerate. For the case of the Hamiltonian, these degeneracies are referred to asmultiplets. Thus, a nondegenerate subspace associated with a state of zero angularmomentum is referred to as a singlet, a doubly-degenerate state associated with atwo-fold degenerate j = 1=2 state is a doublet, and a three-fold degenerate stateassociated with angular momentum j = 1 is referred to as a triplet.
5.10 Addition of Angular Momenta
Let S1 and S2 be two quantum mechanical state spaces associated with angular mo-mentum ~J1 an ~J2; respectively. Let fj¿1; j1;m1ig denote the basis vectors of a standardrepresentation for S1, which is decomposable into corresponding irreducible subspacesS1(¿1; j1), and denote by fj¿2; j2;m2ig the basis vectors of a standard representation forS2; decomposable into irreducible subspaces S2(¿2; j2).
The combined quantum system formed from S1 and S2 is an element of the directproduct space
S = S1 S2: (5.217)
The direct product states
j¿1; j1;m1; ¿2; j2;m2i = j¿1; j1;m1ij¿2; j2;m2i (5.218)
form an orthonormal basis for S. As we will see, however, these direct product statesdo not de…ne a standard representation for S. Indeed, for this combined space the totalangular momentum vector is the sum
~J = ~J1 + ~J2 (5.219)
of those assocated with each “factor space”. What this means is that the rotation op-erators for the combined space are just the products of the rotation operators for eachindividual space
Uu(®) = U(1)u (®)U
(2)u (®)
= e¡i®~J1¢ue¡i®~J2¢u = e¡i®(~J1+~J2)¢u (5.220)
e¡i®~J¢u = e¡i®(~J1+~J2)¢u (5.221)
so that the generators of rotations for each factor space simply add. At this point, wemake no speci…c identi…cation of the nature of the two subspaces involved. Accordingly,the results that we will derive will apply equally well to the description of two spinlessparticles (for which ~J = ~L = ~L1+ ~L2 is the total orbital angular momentum of the pair),to the description of a single particle with spin (for which ~J = ~L + ~S is the sum of theorbital and spin angular momenta of the particle), or even to a collection of particles(where ~J = ~L+ ~S is again the sum of the orbital and spin angular momentum, but wherenow the latter represent the corresponding orbital and spin angular momenta ~L =
P®~L®
and ~S =P®~S® for the entire collection of particles).
194 Angular Momentum and Rotations
In either of these cases the problem of interest is to construct a standard repre-sentation j¿ ; j;mi of common eigenstates of J2 and Jz associated with the total angularmomentum vector ~J of the system as linear combinations of the direct product statesj¿1; j1;m1; ¿2; j2;m2i. As it turns out, the latter are eigenstates of
Jz = ( ~J1 + ~J2) ¢ z= J1z + J2z (5.222)
since they are individually eigenstates of J1z and J2z; i.e.,
Jzj¿1; j1;m1; ¿2; j2;m2i = (J1z + J2z)j¿1; j1;m1; ¿2; j2;m2i= (m1 +m2)j¿1; j1;m1; ¿2; j2;m2i= mj¿1; j1;m1; ¿2; j2;m2i (5.223)
where m = m1 +m2: The problem is that these direct product states are generally noteigenstates of
J2 = ( ~J1 + ~J2) ¢ ( ~J1 + ~J2)
= J21 + J22 + 2
~J1 ¢ ~J2 (5.224)
because, although they are eigenstates of J21 and J22 ; they are not eigenstates of
~J1 ¢ ~J2 =Xi
J1iJ2i (5.225)
due to the presence in this latter expression of operator components of ~J1 and ~J2 perpen-dicular to the z axis.
Using the language of invariant subspaces, another way of expressing the problemat hand is as follows: determine how the direct product space S = S1 S2 can bedecomposed into its own irreducible invariant subspaces S(¿ ; j). This way of thinkingabout the problem actually leads to a simpli…cation. We note that since S1 and S2 caneach be written as a direct sum
S1 =X¿1;j1
S1(¿1; j1) S2 =X¿2;j2
S2(¿2; j2) (5.226)
of rotationally invariant subspaces, the direct product of S1 and S2 can also be writttenas a direct sum
S = S1 S2=
X¿1;j1;¿2;j2
S1(¿1; j1) S2(¿2; j2)
=X
¿1;j1;¿2;j2
S(¿1; ¿2; j1; j2) (5.227)
of direct product subspaces S(¿1; ¿2; j1; j2) = S1(¿1; j1) S2(¿2; j2).Now, because S1(¿1; j1) and S2(¿2; j2) are rotationally invariant, so is their di-
rect product, i.e., any vector j¿1j1m1; ¿2j2m2i in this space will be take by an arbi-trary rotation onto the direct product of two other vectors, one from S1(¿1; j1) andone from S2(¿2; j2); it will remain inside S(¿1; ¿2; j1; j2): On the other hand, althoughS(¿1; ¿2; j1; j2) is rotationally invariant there is no reason to expect it that it is also irre-ducible. However, in decomposing S into irreducible invariant subspaces, we can use the
Addition of Angular Momenta 195
fact that we already have a natural decomposition of that space into smaller invariantsubspaces. To completely reduce the space S we just need to break these smaller partsinto even smaller irreducible parts.
Within any such space S(¿1; ¿2; j1; j2) the values of ¿1; ¿2; j1; and j2 are …xed.Thus we can simplify our notation in accord with the simpler problem at hand whichcan be stated thusly: …nd the irreducible invariant subspaces of a direct product spaceS(¿1; ¿2; j1; j2) = S1(¿1; j1) S2(¿2; j2) with …xed values of j1 and j2: While working inthis subspace we suppress any constant labels, and so denote by
S(j1; j2) = S(¿1; ¿2; j1; j2) = S1(¿1; j1) S2(¿2; j2) (5.228)
the subspace of interest and by
jm1;m2i = j¿1; j1;m1; ¿2; j2;m2i (5.229)
the original direct product states within this subspace. These latter are eigenvectors ofJ21 and J
22 with eigenvalues j1(j1+1) and j2(j2+1); respectively, and of J1z and J2z with
eigenvalues m1 and m2. We will denote the sought-after common eigenstates of J2 andJz in this subspace by the vectors
jj;mi = j¿1; j1; ¿2; j2; j;mi (5.230)
which are to be formed as linear combinations of the states jm1;m2i.With this notation we now proceed to prove the main result, referred to as
The addition theorem: The (2j1 + 1)(2j2 + 1) dimensional space S(j1; j2)contains exactly one irreducible subspace S(j) for each value of j in the se-quence
j = j1 + j2; j1 + j2 ¡ 1; ¢ ¢ ¢ ; jj1 ¡ j2j: (5.231)
In other words, the subspace S(j1; j2) can be reduced into a direct sum
S(j1; j2) = S(j1 + j2)© S(j1 + j2 ¡ 1)© ¢ ¢ ¢S(jj1 ¡ j2j) (5.232)
of irreducible invariant subspaces of the rotation group, where each space S(j)is spanned by 2j + 1 basis vectors jj;mi.
To prove this result we begin with a few general observations, and then followup with what is essentially a proof-by-construction. First, we note that since the spaceS(j1; j2) contains states jm1;m2i with
j1 ¸ m1 ¸ ¡j1 and j2 ¸m2 ¸ ¡j2 (5.233)
the corresponding eigenvalues m = m1 +m2 of Jz within this subspace can only take onvalues in the range
j1 + j2 ¸ m ¸ ¡(j1 + j2): (5.234)
This implies that the eigenvalues of J2 must, themselves be labeled by values of j satisfyingthe bound
j1 + j2 ¸ j: (5.235)
Moreover, it is not hard to see that the sum of m1 and m2 will result in integral valuesof m if m1 and m2are both integral or both half-integral and will result in half-integral
196 Angular Momentum and Rotations
values of m if one is integral and the other half-integral. Since the integral character ofm1 and m2 is determined by the character of j1 and j2; we deduce that
j =
8<: integral if j1; j2 are both integral or both half-integral
half-integral otherwise.(5.236)
With these preliminary observations out of the way, we now proceed to observethat in the subspace S(j1; j2) there is only one direct product state jm1;m2i in which thevalue of m = m1 +m2 takes on its largest value of j1 + j2; namely that vector in whichm1 and m2 both individually take on their largest values, j1 and j2. This fact, we assert,implies that the vector, jm1;m2i = jj1; j2i must be also be an eigenvector of J2 and Jzwith angular momentum (j;m) = (j1 + j2; j1 + j2): In other words, it is that vector ofan irreducible subspace S(j) with j = j1 + j2 having the maximum possible componentof angular momentum along the z axis consistent with that value of j. To prove thisassertion, we note that if this were not the case, we could act on this vector with theraising operator
J+ = J1+ + J2+ (5.237)
and produce an eigenstate of Jz with eigenvalue m = j1 + j2 + 1. This state would haveto be in S(j1; j2) because the latter is invariant under the action of the components of~J . But no vector exists in this space with m larger than j1 + j2: Thus, when J+ acts onjj1; j2i it must take it onto the null vector. The only states having this property are thoseof the form jj;mi with j =m, which proves the assertion.
Since there is only one such state in S(j1; j2) with this value of m; moreover,there can be only one irreducible subspace S(j) with j = j1 + j2 (in general there wouldbe one such vector starting the sequence of basis vectors for each such subspace). Thuswe identify
jj1 + j2; j1 + j2i = jj1; j2i;where the left side of this expression indicates the jj;mi state, the right side indicates theoriginal direct product state jm1;m2i: The remaining basis states jj;mi in this irreduciblespace S(j) with j = j1+ j2 can now, in principle, be produced by repeated application ofthe lowering operator
J¡ = J1¡ + J2¡: (5.238)
For example, we note that, in the jj;mi representation,
J¡jj1 + j2; j1 + j2i =
p(j1 + j2)(j1 + j2 + 1)¡ (j1 + j2)(j1 + j2 ¡ 1) jj1 + j2; j1 + j2 ¡ 1i
=p2(j1 + j2) jj1 + j2; j1 + j2 ¡ 1i: (5.239)
But J¡ = J1¡+J2¡; so this same expression can be written in the jm1;m2i representation,after some manipulation, as
(J1¡ + J2¡)jj1; j2i =p2j1 jj1 ¡ 1; j2i+
p2j2 jj1; j2 ¡ 1i: (5.240)
Equating these last two results then gives
jj1 + j2; j1 + j2 ¡ 1i =s
j2j1 + j2
jj1; j2 ¡ 1i+s
j1j1 + j2
jj1 ¡ 1; j2i: (5.241)
Addition of Angular Momenta 197
This procedure can obviously be repeated for the remaining basis vectors with this valueof j.
We now proced to essentially repeat the argument, by noticing that there areexactly two direct product states jm1;m2i in which the value of m = m1 +m2 takes thenext largest value possible, i.e., m = j1 + j2 ¡ 1; namely, the states
jm1;m2i = jj1; j2 ¡ 1ijm1;m2i = jj1 ¡ 1; j2i: (5.242)
From these two orthogonal states we can produce any eigenvectors of Jz in S(j1; j2)having eigenvalue m = j1 + j2 ¡ 1: In particular, we can form the state (5.241), whichis an eigenvector of J2 with j = j1 + j2: But we can also produce from these two directproduct states a vector orthogonal to (5.241), e.g., the vectors
j2j1 + j2
jj1; j2 ¡ 1i ¡s
j1j1 + j2
jj1 ¡ 1; j2i: (5.243)
Analogous to our previous argument we argue that this latter state must be an eigenstateof J 2 and Jz with angular momentum (j;m) = (j1 + j2 ¡ 1; j1 + j2 ¡ 1): In other words,it is that vector of an irreducible subspace S(j) with j = j1+ j2¡ 1 having the maximumpossible component of angular momentum along the z axis consistent with that value ofj. To prove this assertion, assume it were not the case. We could then act both on thisvector and on (5.241) with the raising operator and produce in S(j1; j2) two orthogonaleigenstates of Jz with eigenvalue m = j1 + j2 (since, as we have seen the raising andlowering operators preserve the orthogonality of such sequences). But there is only onesuch state with this value of m; and it is obtained by applying the raising operator to(5.241). Thus, application of J+ to (5.243) must take it onto the null vector, and henceit must be a state of the type asserted. Since there are no other orthogonal states of thistype that can be constructed, we deduce that there is exactly one irreducible invariantsubspace S(j) with j = j1 + j2 ¡ 1; and we identify (5.243) with the state heading thesequence of basis vectors for that space. As before, the remaining basis vectors jj;mi forthis value of j can then be generated by applying the lowering operator J¡ = J1¡ + J2¡to (5.243).
The next steps, we hope, are clear: repeat this procedure until we run out ofvectors. The basic idea is that as we move down through each value of j we alwaysencounter just enough linearly-independent direct product states with a given value of mto form by linear combination the basis vectors associated with those irreducible spacesalready generated with higher values of j; as well as one additional vector which is to beconstructed orthogonal to those from the other irreducible spaces. This remaining stateforms the beginning vector for a new sequence of 2j +1 basis vectors associated with thepresent value of j.
Thus, e.g., for small enough n; we …nd that there are exactly n+1 direct productstates jm1;m2i in which the value of m = m1 + m2 takes the value m = j1 + j2 ¡ n;namely, the states with
m1 = m2 =j1 ¡ n j2j1 ¡ n+ 1 j2 ¡ 1...
...j1 j2 ¡ n
: (5.244)
If, at this stage, there has been exactly one irreducible invariant subspace S(j) for allvalues of j greater than j1 + j2 ¡ n; than we can form from these n + 1 vectors those
198 Angular Momentum and Rotations
n vectors having this m value that have already been obtained using J¡ from spaceswith higher values of j. We can then form exactly one additional vector orthogonal tothese (e.g., by the Gram-Schmidt procedure) which cannot be associated with any of thesubspaces S(j) already constructed and so, by elimination, must be associated with theone existing subspace S(j) having j = j1 + j2 ¡ n: This state must, moreover, be thatstate of this space in which m = j, (if it were not we could use the raising operator toproduce n+ 1 orthogonal states with m having the next higher value which exceeds thenumber of orthogonal states of this type). Thus, the remaining basis vectors of this spacecan be constructed using the lowering operator on this state. We have inductively shown,therefore, that if there is exactly one irreducible invariant subspace S(j) for all values ofj greater than j1 + j2 ¡ n; then there is exactly one such subspace for j = j1 + j2 ¡ n.This argument proceeds until we reach a value of n for which there are not n + 1 basisvectors with the required value of m. From the table above, we see that this occurs whenthe values of m1 or m2 exceed there natural lower bounds of ¡j1 and ¡j2: Conversely,the induction proof holds for all values of n such that
j1 ¡ n ¸ ¡j1 and j2 ¡ n ¸ ¡j2 (5.245)
or2j1 ¸ n and 2j2 ¸ n (5.246)
With j = j1 + j2 ¡ n; this implies that
j ¸ j2 ¡ j1 and j ¸ j1 ¡ j2 (5.247)
or more simplyj ¸ jj1 ¡ j2j: (5.248)
Thus, the arguments above prove the basic statement of the addition theorem,namely that there exists in S(j1; j2) exactly one space S(j) for values of j starting atj1 + j2 and stepping down one unit at a time to the value of jj1 ¡ j2j: Moreover, themethod of proof contains an outline of the basic procedure used to actually constructthe subspaces of interest. Of course there remains the logical possibility that there existother irreducible subspaces in S(j1; j2) that are simply not accessible using the procedureoutlined. It is straightforward to show that this is not the case, however, by simplycounting the number of vectors produced through the procedure outlined. Indeed, foreach value of j = j1 + j2; ¢ ¢ ¢ ; jj1 ¡ j2j the procedure outlined above generates 2j + 1basis vectors. The total number of such basis vectors is then represented by the readilycomputable sum
j1+j2Xj=jj1¡j2j
(2j + 1) = (2j1 + 1)(2j2 + 1) (5.249)
showing that they are su¢cient in number to generate a space of dimension equal to theoriginal.
Thus, the states jj;mi = j¿1; j1; ¿2; j2; j;mi formed in this way comprise an or-thonormal basis for the subspace S(j1; j2): Implicitly, therefore, there is within this sub-space a unitary transformation between the original direct product states jm1;m2i =j¿1; j1; ¿2; j2;m1;m2i and the basis states jj;mi assocated with the total angular momen-tum ~J . By an appropriate choice of phase, the expansion coe¢cients (i.e., the matrixelements of the unitary transformation between these two sets) can be chosen indepen-dent of ¿ and ¿ 0 and dependent only on the values j;m; j1; j2;m1; and m2: Thus, e.g., the
Addition of Angular Momenta 199
new basis states can be written as linear combinations of the direct product states in theusual way, i.e.,
jj;mi =j1X
m1=¡j1
j2Xm2=¡j2
jj1; j2;m1;m2ihj1; j2;m1;m2jj;mi (5.250)
where we have included the quantum numbers j1 and j2 in the expansion to explicitlyindicate the subspaces that are being combined. Similarly, the direct product states canbe written as linear combinations of the new basis states jj;mi; i.e.,
jj1; j2;m1;m2i =j1+j2X
j=jj1¡j2j
jXm=¡j
jj;mihj;mjj1; j2;m1;m2i: (5.251)
These expansions are completely determined once we know the corresponding expansioncoe¢cients hj1; j2;m1;m2jj;mi; which are referred to as Clebsch-Gordon coe¢cients (orCG coe¢cients). Di¤erent authors denote these expansion coe¢cients in di¤erent ways,e.g.,
Cj1;j2;m1;m2
j;m = hj;mjj1; j2;m1;m2i: (5.252)
It is straightforward, using the procedure outlined above, to generate the CG coe¢cientsfor given values of j1; j2; and j. They obey certain properties that follow from theirde…nition and from the way in which they are constructed. We enumerate some of theseproperties below.
1. Restrictions on j and m - It is clear from the proof of the addition theoremdetailed above that the CG coe¢cient hj1; j2;m1;m2jj;mi must vanish unless thetwo states in the innner product have the same z component of total angular mo-mentum. In addition, we must have the value of total j on the right lie within therange produced by the angular momenta j1 and j2. Thus we have the restriction
hj1; j2;m1;m2jj;mi = 0 unless m =m1 +m2
hj1; j2;m1;m2jj;mi = 0 unless j1 + j2 ¸ j ¸ jj1 ¡ j2j: (5.253)The restriction on j is referred to as the triangle inequality, since it is equivalentto the condition that the positive numbers j; j1; and j2 be able to represent thelengths of the three sides of some triangle. As such, it is easily shown to apply toany permutation of these three numbers, i.e., its validity also implies that j1 + j ¸j2 ¸ jj1 ¡ jj and that j + j2 ¸ j1 ¸ jj ¡ j2j.
2. Phase convention - From the process outlined above, the only ambiguity involvedin constructing the states jj;mi from the direct product states jm1;m2i is at thepoint where we construct the maximally aligned vector jj; ji for each subspace S(j):This vector can always be constructed orthogonal to the states with the same valueof m associated with higher values of j; but the phase of the state so constructedcan, in principle, take any value. To unambiguously specify the CG coe¢cients,therefore, this phase must be unambiguously speci…ed. This is done by de…ning thephase of this state relative to a particular direct product state. In particular, wede…ne the CG coe¢cients so that the coe¢cient
hj1; j2; j1; j ¡ j1jj; ji = hj; jjj1; j2; j1; j ¡ j1i ¸ 0 (5.254)
is real and positive. Since the remaining states jj;mi are constructed from jj; jiusing the lowering operator this choice makes all of the CG coe¢cients real
hj1; j2;m1;m2jj;mi = hj;mjj1; j2;m1;m2i; (5.255)
200 Angular Momentum and Rotations
although not necessarily positive (one can show, for example, that the sign ofhj1; j2;m1;m2jj; ji is (¡1)j1¡m1 .)
3. Orthogonality and completeness relations - Being eigenstates of Hermitianoperators the two sets of states fjj;mig and fjm1;m2ig each form an ONB for thesubspace S(j1; j2): Orthonormality implies that
hj1j2m1m2jj1j2m01m
02i = ±m1;m0
1±m2;m0
2(5.256)
hj;mjj0;m0i = ±j;j0±m;m0 (5.257)
While completeness of each set within this subspace implies that
j1+j2Xj=jj1¡j2j
jXm=¡j
jj;mihj;mj = 1 within S(j1; j2) (5.258)
j1Xm1=¡j1
j2Xm2=¡j2
jj1; j2;m1;m2ihj1; j2;m1;m2j = 1 within S(j1; j2) (5.259)
inserting the completeness relations into the orthonormality relations gives corre-sponding orthonormality conditions for the CG coe¢cients, i.e.,
j1+j2Xj=jj1¡j2j
jXm=¡j
hj1j2m1m2jj;mihj;mjj1j2m01m
02i = ±m1;m0
1(5.260)
j1Xm1=¡j1
j2Xm2=¡j2
hj;mjj1; j2;m1;m2ihj1; j2;m1;m2jj0;m0i = ±j;j0±m;m0 : (5.261)
4. Recursion relation - The states jj;mi in each irreducible subspace S(j) are formedfrom the state jj; ji by application of the lowering operator. It is possible, as aresult, to use the lowering operator to obtain recursion relations for the Clebsch-Gordon coe¢cients associated with …xed values of j; j1; and j2. To develop theserelations we consider the matrix element of J§ between the states jj;mi and thestates jj1; j2;m1;m2i; i.e., we consider
hj;mjJ§jj1; j2;m1;m2i = hj;mj(J1§ + J2§)jj1; j2;m1;m2i (5.262)
On the left hand side of this expression we let J§ act on the bra hj;mj. Since thisis the adjoint of J¨jj;mi the role of the raising and lowering operators is reversed,i.e.,
hj;mjJ§ =pj(j + 1)¡m(m¨ 1) hj;m¨ 1j: (5.263)
Substituting this in above and letting J1§ and J2§ act to the right we obtain therelations p
j(j + 1)¡m(m¨ 1) hj;m¨ 1jj1; j2;m1;m2i
=pj1(j1 + 1)¡m1(m1 § 1) hj;mjj1; j2;m1 § 1;m2i
+pj2(j2 + 1)¡m2(m2 § 1) hj;mjj1; j2;m1;m2 § 1i: (5.264)
These relations allow all CG coe¢cients for …xed j; j1; and j2; to be obtained froma single one, e.g., hj1; j2; j1; j ¡ j1jj; ji.
Addition of Angular Momenta 201
5. Clebsch-Gordon series - As a …nal property of the CG coe¢cients we derive arelation that follows from the fact that the space S(j1; j2) is formed from the directproduct of irreducible invariant subspaces S1(j1) and S2(j2). In particular, we knowthat in the space S1(j1) the basis vectors jj1;m1i transform under rotations intolinear combinations of the themselves according to the relation
U(1)R jj1m1i =
j1Xm01=¡j1
jj1;m01iR(j1)m0
1;m1; (5.265)
where R(j1)m01;m1
is the rotation matrix associated with an irreducible invariant sub-space with angular momentum j1: Similarly, for the states jj2;m2i of S2(j2) we havethe relation
U(2)R jj2m2i =
j2Xm02=¡j2
jj2;m02iR(j2)m0
2;m2: (5.266)
It follows that in the combined space the direct product states jj1; j2;m1;m2i trans-form under rotations as follows
URjj1; j2;m1;m2i = U(1)R U
(2)R jj1; j2;m1;m2i
=Xm01;m
02
jj1; j2;m01;m
02iR(j1)m0
1;m1R(j2)m02;m2
: (5.267)
On the other hand, we can also express the states jj1; j2;m1;m2i in terms of thestates jj;mi; i.e.,
URjj1; j2;m1;m2i =Xj;m
URjj;mihj;mjj1; j2;m1;m2i: (5.268)
But the states jj;mi are the basis vectors of an irreducible invariant subspace S(j)of the combined subspace, and so tranform accordingly,
URjj;mi =Xm0jj;m0iR(j)m0;m: (5.269)
Thus, we deduce that
URjj1; j2;m1;m2i =Xj;m;m0
jj;m0iR(j)m0;mhj;mjj1; j2;m1;m2i (5.270)
=Xm01;m
02
Xj;m;m0
jj1; j2;m01;m
02ihj1; j2;m0
1;m02jj;m0iR(j)m0;mhj;mjj1; j2;m1;m2i
(5.271)where in the second line we have transformed the states jj;m0i back to the directproduct representation. Comparing coe¢cients in (5.267) and (5.271), we deduce arelation between matrix elements of the rotation matrices for di¤erent values of j;namely,
R(j1)m01;m1
R(j2)m02;m2
=Xj;m;m0
hj1; j2;m01;m
02jj;m0iR(j)m0;mhj;mjj1; j2;m1;m2i (5.272)
which is referred to as the Clebsch-Gordon series. Note that in this last expressionthe CG coe¢cients actually allow the sum overm andm0 to both collapse to a single
202 Angular Momentum and Rotations
term with m =m1 +m2 and m0 = m01 +m
02; making it equivalent to
R(j1)m01;m1
R(j2)m02;m2
=
j1+j2Xj=jj1¡j2j
hj1; j2;m01;m
02jj;m0
1+m02iR(j)m0
1+m02;m1+m2
hj;m1+m2jj1; j2;m1;m2i
(5.273)
5.11 Reducible and Irreducible Tensor Operators
We have seen that it is possible to classify observables of a system in terms of theirtransformation properties. Thus, scalar observables are, by de…nition, invariant underrotations. The components of a vector observable, on the other hand, transform intowell de…ned linear combinations of one another under an arbitrary rotation. As it turnsout, these two examples constitute special cases of a more general classi…cation schemeinvolving the concept of tensor operators. By de…nition, a collection of n operatorsfQ1; Q2; ¢ ¢ ¢ ; Qng comprise an n-component rotational tensor operatorQ if they transformunder rotations into linear combinations of each other, i.e., if for each rotation R thereare a set of coe¢cients Dji(R) such that
R [Qi] = URQiU+r =
Xj
QjDji(R): (5.274)
It is straightforward to show that under these circumstances the matrices D(R) withmatrix elements Dji(R) form a representation for the rotation group.
As an example, we note that the Cartesian components fVx; Vy; Vzg of a vectoroperator ~V form the components of a 3-component tensor V. Indeed, under an arbitraryrotation, the operator Vu = ~V ¢ u is transformed into
R [Vu] = Vu0 = ~V ¢ u0 (5.275)
where u0 = ARu indicates the direction obtained by performing the rotation R on thevector u. Thus,
u0i =Xj
Aijuj : (5.276)
If u corresponds to the Cartesian unit vector xk; then uj = ±j;k and u0i = Aik; so that
R [Vk] = ~V ¢ x0k =Xi
ViAik (5.277)
which shows that the components of ~V transform into linear combinations of one anotherunder rotations, with coe¢cients given by the 3£ 3 rotation matrices AR.
As a second example, if ~V and ~W are vector operators and Q is a scalar opera-tor, then the operators fQ;Vx; Vy; Vz;Wx;Wy;Wzg form a seven component tensor, sinceunder an arbitrary rotation R they are taken onto
Q ! Q+Xj
0 ¢ Vj +Xj
0 ¢Wj (5.278)
Vi !Xj
VjAji +Xj
0 ¢Wj + 0 ¢Q (5.279)
Wi !Xj
WjAji +Xj
0 ¢ Vj + 0 ¢Q (5.280)
Reducible and Irreducible Tensor Operators 203
which satis…es the de…nition. Clearly in this case, however, the set of seven componentscan be partitioned into 3 separate sets of operators fQg; fVx; Vy; Vzg; and fWx;Wy;Wzg;which independently transform into linear combinations of one another, i.e., the tensorcan be reduced into two 3-component tensors and one tensor having just one component(obviously any scalar operator constitutes a one-component tensor). Note also that inthis circumstance the matrices D(R) governing the transformation of these operators isblock diagonal, i.e., it has the form
D(R) =
0BBBBBBBB@
1 0@ AR
1A 0@ AR
1A
1CCCCCCCCA: (5.281)
Thus this representation of the rotation group is really a combination of three separaterepresentations, one of which is 1 dimensional and two of which are 3 dimensional.
This leads us to the idea of irreducible tensors. A tensor operator T is said tobe reducible if its components fT1; T2; ¢ ¢ ¢ ; Tng, or any set of linear combinations thereof,can be partitioned into tensors having a smaller number of components. If a tensor cannotbe so partitioned it is said to be irreducible.
For example, if ~V is a vector operator, the set of operators fVx; Vyg do not forma two component tensor, because a rotation of Vx about the y axis through ¼=2 takes itonto Vz; which cannot be expressed as a linear combination of Vx and Vy: Thus, a vectoroperator cannot be reduced into smaller subsets of operators. It follows that all vectoroperators are irreducible.
It is interesting to note that the language that we are using here to describe thecomponents of tensor operators is clearly very similar to that describing the behavior ofthe basis vectors associated with rotationally invariant subspaces of a quantum mechanicalHilbert space. Indeed, in a certain sense the basic reducibility problem has alread beensolved in the context of combining angular momenta. We are led quite naturally, therefore,to introduce a very useful class of irreducible tensors referred to as spherical tensors.
By de…nition, a collection of 2j+1 operators fTmj jm = ¡j; ¢ ¢ ¢ ; jg form the com-ponents of an irreducible spherical tensor Tj of rank j if they transform under rotationsinto linear combinations of one another in the same way as the basis vectors jj;mi of anirreducible invariant subspace S(j): Speci…cally, this means that under a rotation R; theoperator Tmj is taken onto
R£Tmj¤= URT
mj U
+R =
Xm0Tm
0j R
(j)m0;m (5.282)
where R(j)m0;m represents the rotation matrix associated with an eigenspace of J2 with this
value of j. Since the basis vectors jj;mi transform irreducibly, it is not hard to see thatthe components of spherical tensors of this sort do so as well.
It is not hard to see that, according to this de…nition, a scalar observable Q = Q00is an irreducible spherical tensor of rank zero, i.e., its transformation law
R£Q00¤= Q00 (5.283)
is the same as that of the single basis vector j0; 0i associated with a subspace of zeroangular momentum, for which
R [j0; 0i] = j0; 0i: (5.284)
204 Angular Momentum and Rotations
Similary, a vector operator ~V , which de…nes an irreducible three component tensor, can beviewed as a spherical tensorV1 of rank one. By de…nition, the spherical tensor componentsfV m1 jm = 1; 0;¡1g of a vector operator ~V are given as the following linear combinations
V 11 = ¡(Vx + iVy)p
2V 01 = Vz V ¡11 =
Vx ¡ iVyp2
(5.285)
of its Cartesian components. In this representation we see that the raising and loweringoperators can be expressed in terms of the spherical tensor components of the angularmomentum operator ~J through the relation
J§ = ¨p2J§11 : (5.286)
To see that the spherical components of a vector de…ne an irreducible tensor ofunit rank we must show that they transform appropriately. To this end it su¢ces todemonstrate the transformation properties for any vector operator, since the transfor-mation law will clearly be the same for all vector operators (as it is for the Cartesiancomponents). Consider, then, in the space of a single particle the vector operator Rwhich has the e¤ect in the position representation of multiplying the wavefunction at ~rby the radial unit vector r; i.e.,
Rj~ri = rj~ri = ~r
j~rj j~ri (5.287)
h~rjRjÃi = rÃ(~r) =~rÃ(~r)
j~rj : (5.288)
In the position representation the Cartesian components of this operator can be writtenin spherical coordinates (r; µ; Á) in the usual way, i.e.,
Rx =x
r= cosÁ sin µ (5.289)
Ry =y
r= sinÁ sin µ (5.290)
Rz =z
r= cos µ (5.291)
In this same representation the spherical components of this vector operator take the form
R11 = ¡ 1p2(cosÁ sin µ + i sinÁ sin µ) = ¡ 1p
2eiÁ sin µ (5.292)
R01 = cos µ (5.293)
R¡11 =1p2(cosÁ sin µ ¡ i sinÁ sin µ) = 1p
2e¡iÁ sin µ (5.294)
Aside from an overall constant, these are equivalent to the spherical harmonics of orderone, i.e.,
R11 =
r4¼
3Y 11 (µ; Á) (5.295)
R01 =
r4¼
3Y 01 (µ; Á) (5.296)
R¡11 =
r4¼
3Y ¡11 (µ; Á) (5.297)
Reducible and Irreducible Tensor Operators 205
which, as we have seen, transform as the basis vectors of an irreducible subspace withj = 1
R [Ym1 ] =1X
m0=¡1Y m
01 R
(1)m0;m (5.298)
from which it follows that the spherical components of R transform in the same way, i.e.,
RhRm1
i=
1Xm0=¡1
Rm0
1 R(1)m0;m (5.299)
Thus, this vector operator (and hence all vector operators) de…ne a spherical tensor ofrank one.
As a natural extension of this, it is not hard to see that the spherical harmonicsof a given order l de…ne the components of a tensor operator in (e.g.) the position repre-sentation. Thus, we can de…ne an irreducible tensor operator Yl with 2l+ 1 componentsfY ml jm = ¡l; ¢ ¢ ¢ ; lg which have the following e¤ect in the position representation
h~rjY ml jÃi = Y ml (µ; Á)Ã(r; µ; Á): (5.300)
We note in passing that the components of this tensor operator arise quite naturally inthe multipole expansion of electrostatic and magnetostatic …elds.
The product of two tensors is, itself, generally a tensor. For example, if Tj1and Qj2 represent spherical tensors of rank j1 and j2; respectively, then the set of prod-ucts
©Tm1j1Qm2j2
ªform a (2j1 + 1)(2j2 + 1) component tensor. In general, however, such
a tensor is reducible into tensors of smaller rank. Indeed, since the components Tm1j1;
Qm2j2of each tensor transform as the basis vectors jj1;m1i; jj2;m2i of an irreducible sub-
space S(j1); S(j2) the process of reducing the product of two spherical tensors into irre-ducible components is essentially identical to the process of reducing a direct product spaceS(j1; j2) into its irreducible components. Speci…cally, from the components
©Tm1j1Qm2j2
ªwe can form, for each j = j1+ j2; ¢ ¢ ¢ ; jj1¡ j2j an irreducible tensorWj with components
Wmj =
j1Xm1=¡j1
j2Xm2=¡j2
Tm1j1Qm2j2hj1; j2;m1;m2jj;mi m = ¡j; ¢ ¢ ¢ ; j: (5.301)
To show that these 2j+1 components comprise a spherical tensor of rank j we must showthat they satisfy the approriate transformation law. Consider
R£Wmj
¤= URW
mj U
+R =
Xm1;m2
URTm1j1Qm2j2U+R hj1; j2;m1;m2jj;mi
=Xm1;m2
URTm1j1U+RURQ
m2j2U+R hj1; j2;m1;m2jj;mi
=Xm1;m2
Xm01;m
02
Tm01
j1Qm02
j2R(j1)m01;m1
R(j2)m02;m2
hj1; j2;m1;m2jj;mi (5.302)
206 Angular Momentum and Rotations
Using the Clebsch-Gordon series this last expression can be written
R£Wmj
¤=
Xm1;m2
Xm01;m
02
Xj0;m0;m00
Tm01
j1Qm02
j2hj1; j2;m0
1;m02jj0;m0iR(j0)m0;m00
£hj0;m00jj1j2;m1;m2ihj1; j2;m1;m2jj;mi
=Xm01;m
02
Xj;m0;m00
Tm01
j1Qm02
j2hj1; j2;m0
1;m02jj0;m0iR(j)m0;m00hj0;m00jj;mi
=Xm0
Xm01;m
02
Tm01
j1Qm02
j2hj1; j2;m0
1;m02jj;m0iR(j)m0;m
=Xm0Wm0j R
(j)m0;m (5.303)
which shows that Wmj does indeed transform as the mth component of a tensor of rank
j; and where we have used the orthonormality and completeness relations associated withthe C-G coe¢cients.
5.12 Tensor Commutation Relations
Just as scalar and vector observables obey characteristic commutation relations
[Ji; Q] = 0 (5.304)
[Ji; Vj ] = iXijk
"ijkVk (5.305)
with the components of angular momentum, so do the components of a general sphericaltensor operator. As with scalars and vectors, these commutation relations follow from theway that these operators transform under in…nitesimal rotations. Recall that under anin…nitesimal rotation Uu(±®) = 1¡ i±®Ju; an arbitrary operator Q is transformed into
Q0 = Q¡ i±® [Ju; Q] (5.306)
Thus, the mth component of the spherical tensor Tj is transformed by Uu(±®) into
Ru(±®)[Tmj ] = T
mj ¡ i±® £Ju; Tmj ¤ (5.307)
On the other hand, by de…nition, under any rotation Tmj is transformed into
Ru(±®)[Tmj ] =
jXm0=¡j
Tm0
j R(j)m0;m(u; ±®): (5.308)
But
R(j)m0;m(u; ±®) = hj;m0jUu(±®)jj;mi
= hj;m0j1¡ i±®Jujj;mi= ±m;m0 ¡ i±®hj;m0jJujj;mi (5.309)
which implies that
Ru(±®)[Tmj ] = T
mj ¡ i±®
jXm0=¡j
Tm0
j hj;m0jJujj;mi: (5.310)
The Wigner Eckart Theorem 207
Comparing these we deduce the commutation relations
£Ju; T
mj
¤=
jXm0=¡j
Tm0
j hj;m0jJujj;mi: (5.311)
As special cases of this we have £Jz; T
mj
¤= mTmj :
and £J§; Tmj
¤=pj(j + 1)¡m(m§ 1) Tm§1j : (5.312)
5.13 The Wigner Eckart Theorem
We now prove and important result regarding the matrix elements of tensor operatorsbetween basis states j¿; j;mi of any standard representation. This result, known as theWigner-Eckart theorem, illustrates, in a certain sense, the constraints put on the compo-nents of tensor operators by the transformation laws that they satisfy. Speci…cally, wewill show that the matrix elements of the components of a spherical tensor operator TJbetween basis states j¿ ; j;mi are given by the product
h¿ ; j;mjTMJ j¿ 0; j0;m0i = hj;mjJ; j0;M;m0ih¿; jjjT jj¿ 0j0i (5.313)
of the Clebsch Gordon coe¢cient hj;mjJ; j0;M;m0i and a quantity h¿ ; jjjT jj¿ 0j0i thatis independent of m;M; and m0; referred to as the reduced matrix element. Thus, the“orientational” dependence of the matrix element is completely determined by geometricalconsiderations. This result is not entirely surprising, given that the two quantities on theright hand side of the matrix element, i.e., the TMJ j¿ 0; j0;m0i transform under rotations likea direct product ket of the form jJ;Mi jj0;m0i; while the quantity on the left transformsas a bra of total angular momentum (j;m). The reduced matrix element h¿; jjjT jj¿ 0j0icharacterizes the extent to which the given tensor operator TJ mixes the two subspacesS(¿; j) and S(¿ 0; j0); and is generally di¤erent for each tensor operator.
To prove theWigner-Eckart theorem we will simply show that the matrix elementsof interest obey the same recursion relations as the CG coe¢cients. To this end, we usethe simplifying notation
Cj1;j2;m1;m2
j;m = hj;mjj2; j1;m1;m2i (5.314)
for the CG coe¢cients and denote the matrix elements of interest in a similar fashion,i.e.,
T j1;j2;m1;m2
j;m = h¿; j;mjTm2j1j¿ 0; j2;m2i: (5.315)
This latter quantity is implicitly a function of the labels ¿ ; ¿ 0 , but we will suppress thisdependence until it is needed. We then recall that the CG coe¢cients obey a recursionrelation that is generated by consideration of the matrix elements
hj;mjJ§jj2; j1;m1;m2i = hj;mj(J1§ + J2§)jj2; j1;m1;m2i (5.316)
which leads to the relationpj(j + 1)¡m(m¨ 1)Cj1;j2;m1;m2
j;m¨1 =pj1(j1 + 1)¡m1(m1 § 1) Cj1;j2;m1§1;m2
j;m
+pj2(j2 + 1)¡m2(m2 § 1)Cj1;j2;m1;m2§1
j;m (5.317)
208 Angular Momentum and Rotations
To obtain a similar relation for the matrix elements of Tj1 we consider an “analogous”matrix element
h¿ ; j;mjJ§Tm1j1j¿ 0; j2;m2i =
pj(j + 1)¡m(m¨ 1) h¿; j;m¨ 1jTm1
j1j¿ 0; j2;m2i
=pj(j + 1)¡m(m¨ 1) T j1;j2;m1;m2
j;m¨1 : (5.318)
We can evaluate this in a second way by using the commutation relations satis…ed by J§and Tm1
j1; i.e., we can write
J§Tm1j1
=£J§; Tm1
j1
¤+ Tm1
j1J§
=pj1(j1 + 1)¡m1(m1 § 1) Tm1§1
j1+ Tm1
j1J§ (5.319)
which allows us to express the matrix element above in the form
h¿; j;mjJ§Tm1j1j¿ 0; j2;m2i =
pj1(j1 + 1)¡m1(m1 § 1) h¿ ; j;mjTm1§1
j1j¿ 0; j2;m2i
+h¿ ; j;mjTm1j1J§j¿ 0; j2;m2i (5.320)
which reduces to
h¿ ; j;mjJ§Tm1j1j¿ 0; j2;m2i =
pj1(j1 + 1)¡m1(m1 § 1) T j1;j2;m1§1;m2
j;m
+pj2(j2 + 1)¡m2(m2 § 1) T j1;j2;m1;m2§1
j;m (5.321)
Comparing the two expressions for h¿; j;mjJ§Tm1j1j¿ 0; j2;m2i we deduce the recursion
relationpj(j + 1)¡m(m¨ 1) T j1;j2;m1;m2
j;m¨1 =pj1(j1 + 1)¡m1(m1 § 1) T j1;j2;m1§1;m2
j;m
+pj2(j2 + 1)¡m2(m2 § 1) T j1;j2;m1;m2§1
j;m (5.322)
which is precisely the same as that obeyed by the Clebsch-Gordon coe¢cients Cj1;j2;m1;m2
j;m :The two sets of number, for given values of j; j1; and j2; must be proportional to oneanother. Introducing the reduced matrix element h¿ ; jjjT jj¿ 0j0i as the constant of propor-tionality, we deduce that
T j1;j2;m1;m2
j;m = h¿ ; jjjT jj¿ 0j0iCj1;j2;m1;m2
j;m (5.323)
which becomes after a little rearrangement
h¿ ; j;mjTMJ j¿ 0; j0;m0i = hj;mjJ; j0;M;m0ih¿ ; jjjT jj¿ 0j0i: (5.324)
This theorem is very useful because it leads automatically to certain selectionrules. Indeed, because of the CG coe¢cient on the right hand side we see that the matrixelement of TMJ between two states of this type vanishes unless
¢m =m¡m0 =M (5.325)
j0 + J ¸ j ¸ jj0 ¡ J j: (5.326)
Thus, for example we see that the matrix elements of a scalar operator Q00 vanishunless ¢m = 0 and ¢j = j ¡ j0 = 0: Thus, scalar operators cannot change the angularmomentum of any states that they act upon. (They are often said to carry no angularmomentum, in contrast to tensor operators of higher rank, which can and do change the
The Wigner Eckart Theorem 209
angular momentum of the states that they act upon.) Thus the matrix elements for scalaroperators take the form
h¿ ; j;mjQ00j¿ 0; j0;m0i = Q¿;¿ 0±j;j0±m;m0 : (5.327)
In particular, it follows that within any irreducible subspace S(¿ ; j) the matrix represent-ing any scalar Q00 is just a constant Q¿ times the identity matrix for that space (con…rmingthe rotational invariance of scalar observables within any such subspace), i.e.,
h¿ ; j;mjQ00j¿; j;m0i = Q¿±m;m0 : (5.328)
Application of the Wigner-Eckart theorem to vector operators ~V ; leads to con-sideration of the spherical components fV m1 jm = 0;§1g of such an operator. The corre-sponding matrix elements satisfy the relation
h¿ ; j;mjVM1 j¿ 0; j0;m0i = hj;mj1; j0;M;m0ih¿ ; jjjV jj¿ 0j0i; (5.329)
and vanish unless¢m =M 2 f0;§1g : (5.330)
Similarly, application of the triangle inequality to vector operators leads to the selectionrule
¢j = 0;§1: (5.331)
Thus, vector operators act as though they impart or take away angular momentum j = 1.The matrix elements of a vector operator within any given irreducible space are
proportional to those of any other vector operator, such as the angular momentum oper-ator ~J; whose spherical components satisfy
h¿ ; j;mjJM1 j¿ 0; j0;m0i = hj;mj1; j0;M;m0ih¿ ; jjjJ jj¿ 0j0i; (5.332)
from which it follows that
h¿ ; j;mjVM1 j¿ ; j;m0i = ®(¿ ; j)h¿ ; j;mjJM1 j¿; j;m0i (5.333)
where ®(¿; j) = h¿ ; jjjV jj¿ ; ji=h¿; jjjJ jj¿ ; ji is a constant. Thus, within any subspaceS¿ (j) all vector operators are proportional, we can write
~V = ®~J within S¿ (j): (5.334)
It is a straight forward exercise to compute the constant of proportionality in terms ofthe scalar observable ~J ¢ ~V ; the result being what is referred to as the projection theorem,i.e.,
~V =h ~J ¢ ~V ij(j + 1)
~J within S¿ (j); (5.335)
where the mean value h ~J ¢ ~V i; being a scalar with respect to rotation can be taken withrespect to any state in the subspace S¿ (j):
In a similar manner one …nds that the nonzero matrix elements within any irre-ducible subspace are proportional, i.e., for two nonzero tensor operators TJ and WJ ofthe same rank, it follows that, provided h¿ ; jjjW jj¿ji 6= 0;
h¿ ; j;mjTMJ j¿ 0; j0;m0i = ®h¿; j;mjWMJ j¿ 0; j0;m0i (5.336)
where ® = h¿; jjjT jj¿ 0j0i=h¿ ; jjjW jj¿ 0; j0i: Thus, the orientational dependence of the (2j+1)(2J + 1)(2j0 + 1) matrix elements is completely determined by the transformationalproperties of the states and the tensors involved.
210 Angular Momentum and Rotations
Chapter 4BOUND STATES OF A CENTRAL POTENTIAL
4.1 General Considerations
As an application of some of these ideas we briefly investigate the properties of a particle ofmassm subject to a force deriving from a spherically symmetric potential V (r). Obviously,in many of the cases where this problem is of interest, the massm referred to is the reducedmass
m =m1m2
m1 +m2(4.1)
of an interacting two-particle system and the radial coordinate r corresponds to the mag-nitude of the relative position vector ~r = ~r2−~r1, although these details need not concernus here. The corresponding quantum system is governed by a Hamiltonian operator
H =P 2
2m+ V (R) (4.2)
which in the position representation takes the usual form
−~2∇22m
+ V (r), (4.3)
and our goal is information about the bound state solutions |ψi to the energy eigenvalueequation
(H − ε)|ψi = 0, (4.4)
where we assume that V (r) → 0 as r → ∞, so that bound state solutions are identifiedas those square normalizeable solutions
hψ|ψi =Zd3r |ψ(~r)|2 = 1 (4.5)
for which ε ≤ 0.The spherical symmetry of V (r) ( and thus of H) suggest the use of spherical
coordinates for which the identity
∇2 = 1
r
∂2
∂r2r +
1
r2
·∂2
∂θ2+ cot θ
∂
∂θ+
1
sin2 θ
∂2
∂φ2
¸(4.6)
holds for all points except at r = 0, which is a singular point of the transformation. Thislast expression can be written in the form
P 2 = −~2∇2 = −~2
r
∂2
∂r2r +
~2L2
r2(4.7)
where
L2 = −·∂2
∂θ2+ cot θ
∂
∂θ+
1
sin2 θ
∂2
∂φ2
¸(4.8)
124 Bound States of a Central Potential
is the operator, in this representation, associated with the square of the (dimensionless)orbital angular momentum ~L = ~R× ~K. Thus, as in classical mechanics, the kinetic energyH0 separates into a radial part and a rotational part
H0 =P 2r2m
+~2L2
2mr2, (4.9)
where
P 2r = −~21
r
∂2
∂r2r =
·~i
1
r
∂
∂rr
¸2. (4.10)
This last form suggests that the operator
Pr =~i
1
r
∂
∂rr (4.11)
can be viewed as the radial component of the momentum. This is a legitimate inference,and indeed Pr as defined above does correspond to the Hermitian part
Pr =1
2
h~P · R+ R · ~P
i. (4.12)
of the operator ~P · R. Some care must be taken, however, because (it can be shown that)Pr is only Hermitian on the space of wave functions ψ(~r) such that limr→0 rψ(~r) = 0,while the eigenfunctions of Pr all diverge at r = 0 as r−1. Thus, Pr is an example of aHermitian operator that is not an observable. None of this is too important for the taskat hand, however. Indeed, with the aforementioned formulae we can write the energyeigenvalue equation of interest in the form½
− ~2
2m
1
r
∂2
∂r2r +
~2L2
2mr2+ V (r)− ε
¾ψ(~r) = 0 (4.13)
which is valid at all points except r = 0.We now make the observation that L2 and Lz = −i∂/∂φ (or any other component
of ~L ) commute with the each other and with any function of r or Pr. It is obvious,therefore, that they commute with H. Thus, because£
H,L2¤=£L2, Lz
¤= [H,Lz] = 0 (4.14)
we know that there exists a basis of eigenstates common to the three operators H,L2, andLz. These states, which we will denote by |n, l,mi are characterized by their associatedeigenvalues
H|n, l,mi = εn,l|n, l,mi (4.15)
L2|n, l,mi = l(l + 1)|n, l,mi (4.16)
Lz|n, l,mi = m|n, l,mi (4.17)
and we denote byψn,l,m(~r) = h~r|n, l,mi (4.18)
the wavefunctions in the position representation associated with these states. Note thatthe eigenvalues of H are labeled by the principle quantum number n and by the totalangular momentum quantum number `, but not by the azimuthal quantum number m,thus reflecting the necessary rotational degeneracy associated with the scalar operator H.
General Considerations 125
It is also clear from our earlier discussions that the eigenfunctions of L2 and Lz can bewritten in the general form
ψn,l,m(~r) = Fn,l(r)Yml (θ,φ) (4.19)
involving the spherical harmonics, with ` = 0, 1, 2, · · · , and m = −`, · · · ,+`. Substitutionof this assumed form into the energy eigenvalue equation results in the following ordinarydifferential equation
− ~2
2m
1
r
d2
dr2r Fn,l +
µ`(`+ 1)~2
2mr2+ V (r)− εn,l
¶Fn,l = 0 (4.20)
for the radial functions Fn,l(r). This equation is independent of the eigenvalue m ofLz, confirming our labeling of the eigenfunctions based upon the expected rotationaldegeneracy of the system. Thus, we can anticipate that each energy eigenvalue εn,l willbe 2` + 1 fold degenerate due to the rotational invariance of H. Since the differentialequation does, generally, depend upon the quantum number `, we can expect a differentseries of eigenvalues εn,l for each value of l. Additional accidental degeneracies may arise,however, (and in fact do arise when V (r) = kr2, which corresponds to the 3D harmonicoscillator, and when V (r) = −k/r, which corresponds to the Coulomb potential.)
As a further simplification it is customary to introduce a new function
φn,l(r) = rFn,l(r) (4.21)
which obeys the equation
− ~2
2mφ00n,l +
·`(`+ 1)~2
2mr2+ V (r)− εn,l
¸φn,l = 0. (4.22)
This latter equation is of the precise form
− ~2
2mφ00n,l + [Veff (r)− εn,l]φn,l = 0.
obeyed by a particle moving in a 1-dimensional effective potential
Veff (r) =`(`+ 1)~2
2mr2+ V (r) (4.23)
consisting of the so-called “centrigugal barrier” `(`+1)~2/2mr2 in addition to the centralpotential experienced by the particle. Note that the wavefunction is defined only for r > 0.Moreover, the normalization conditionZ
r2dr dΩ¯ψn,l,m
¯2= 1 (4.24)
leads, along with the standard normalization condition for the spherical harmonicsZdΩ |Y m` |2 = 1 (4.25)
to a normalization conditionZ ∞0
dr r2 |Fn,l(r)|2 =Z ∞0
dr¯φn,l(r)
¯2= 1 (4.26)
126 Bound States of a Central Potential
for the functions φn,l which make them correspond to a particle whose wave function isrestricted to the positive real axis. As we will see, the boundary condition obeyed by thesefunctions for most cases of interest corresponds to one in which φn,l → 0 as r → 0, asthough there were an infinite barrier at the origin confining to the particle to the regionr > 0.
To see this, we consider the general properties of the solution to Eq.(4.22) forsmall values of r. First, we rewrite the equation in the form
φ00n,l −·`(`+ 1)
r2+ v(r) + kn,l
¸φn,l = 0 (4.27)
where
v(r) =2mV (r)
~2k2n,l =
2m |εn,l|~2
(4.28)
and we are assuming the εn,l < 0, as is appropriate to bound state solutions. We furtherassume the following:
1. V (r) is bounded except near r = 0.
2. |V (r)| ≤M/r as r → 0 for some positive constantM. In other words, the magnitudeof V diverges at the origin no more quickly than the Coulomb potential.
3. With these constraints, we now assume a solution near the origin which is of theform of a power law in r, i.e., we assume that
φn,l(r) ∼ Crs+1 r → 0 (4.29)
for some positive constant s. This assumed form corresponds to the actual radialwave function having the behavior Fn,l(r) ∼ Crs as r → 0.
If we now substitute these limiting forms into the energy eigenvalue equation weobtain the following algebraic relation
s(s+ 1)rs−1 − `(`+ 1)rs−1 − v(r)rs+1 +−k2n,lrs+1 = 0. (4.30)
Now as r → 0, the last two terms are at most of order rs and become negligible comparedto terms of order rs−1. Thus, for this form to be consistent at small r we must haves(s+ 1) = `(`+ 1), which has two solutions
s = ` s = −(`+ 1). (4.31)
In principle, this gives us two linearly independent solutions to the second order differentialequation. The solution with s = ` gives a solution for which, at small r,
φn,l(r) ∼ Cr`+1 Fn,l(r) ∼ Cr` (4.32)
while the solution with s = −(` + 1) gives a solution which for small r has the limitingbehavior
φn,l(r) ∼C
r`Fn,l(r) ∼ C
r`+1. (4.33)
Of these, the solutions of the second type can be rejected on physical grounds. First, if` > 0, then the solution of the form φ(r) ∼ C/r` is not normalizeable
|C|2Z a
ε
dr
r2`→∞ ε→ 0. (4.34)
Hydrogenic Atoms: The Coulomb Problem 127
In the l = 0 case the situation is a little different. In fact, if l = 0, the kinetic energyoperator acting on any function proportional to 1/r gives a delta function, i.e., for r nearzero,
∇2Fn,0(r) ∼ ∇2·1
r
¸= 4πδ(~r). (4.35)
The presence of this delta function prevents the energy eigenvalue equation from beingsatisfied at the origin, since the potential only has at most algebraic singularities at thatpoint. Thus, the second “solution” for l = 0 is not actually a valid solution to theeigenvalue equation. It is a spurious solution arising from the singular nature of thetransformation to spherical coordinates at r = 0. Thus, in the end we conclude that underthese conditions we obtain exactly one regular radial function
φn,l ∼ Cr`+1 Fn,l ∼ Cr` r → 0 (4.36)
for each value ` = 0, 1, 2, · · · of the orbital angular momentum. Note that this limitingform implies the following “boundary condition”
limr→0
φn,l(r) = 0 (4.37)
for the effective one-dimensional problem, making the wave function act as though therewere an infinite potential barrier at r = 0.
We have gone about as far as we can go without actually specifying the potential.We thus turn to what is certainly the most important application of these ideas to atomicsystems, namely, the Coulomb problem associated with a particle moving in a Coulombicpotential well.
4.2 Hydrogenic Atoms: The Coulomb Problem
We now assume that potential of interest is of the form
V (r) = − 1
4πε0
Zq2
r(4.38)
associated with the Coulomb interaction between a nucleus of charge +Zq and a singleelectron of charge −q. We ignore the spin of the contituents, and rewrite the potential as
V (r) = −Ze2
r(4.39)
by introducing e2 = q2/4πε0. From dimensional arguments it is possible to generateestimates of the energy and length scales associated with bound states of this system. Thisis a useful exercise since it allows us to express the equations of interest in dimensionlessform. These scales of interest arise from the fact that the mean energy
hEi = hT i+ hV i (4.40)
of a bound state depends upon both its kinetic energy
hT i = hP 2i2m
(4.41)
and its potential energy
hV i ∼ −Ze2
a(4.42)
128 Bound States of a Central Potential
where a ∼ hri is a typical distance of the electron from the origin, i.e., its spread a ∼ ∆rabout the nucleus. In any stationary bound state the mean position of the particle mustitself be stationary, which requires that h~P i = 0, and so, using the uncertainty principlewe have the estimate
∆P =phP 2i ∼ ~
∆r=~a
hT i ∼ ~2
2ma2. (4.43)
Thus, an order of magnitude estimate gives
hEi ∼ ~2
2ma2− Ze
2
a. (4.44)
To estimate the ground state energy, therefore, we minimize this with respect to theparameter a. Setting ∂hEi/∂a = 0 we find that
a =1
Z
~2
me2=a0Z
(4.45)
where
a0 =~2
me2∼ 0.51angstrom (4.46)
turns out to be the Bohr radius. Putting this into our expression for the energy gives theestimate
E = −Z2~2
2ma20= −Z2ε0 ∼ −Z2 × 13.6eV (4.47)
where
ε0 =~2
2ma20∼ 13.6eV
In addition, we can obtain an estimate for a typical velocity for the problem through therelation
v =p
m∼ ~ma
= Ze2
~= Zv0 ∼ Z × 2.2× 106m/s. (4.48)
In atomic physics it is useful to introduce a set of units in which e = ~ = me = 1, so thata0 = 1, v0 = 1, and E0 = 1/2. We shall not do that in the present treatment, but we willuse these length and energy scales to appropriately transform the problem. For example,the radial equation for this problem takes the form
~2
2mφ00n,l +
µZe2
r− l(l + 1)~
2
2mr2+En,l
¶φn,l = 0. (4.49)
Introducing a dimensionless position variable ρ = r/a this can be written in the form
~2
2ma2d2φ
dρ2+
µZe2
2a
1
ρ− l(l + 1)~
2
2ma21
ρ2+En,l
¶φ = 0. (4.50)
Dividing this through by E0 = Z2ε0 = ~2/2ma2 this reduces to the form
d2φ
dρ2+
µ2
ρ− l(l + 1)
ρ2− λ2n,l
¶φ = 0 (4.51)
where
λ2n,l = −En,lZ2ε0
≥ 0 (4.52)
Hydrogenic Atoms: The Coulomb Problem 129
is a positive dimensionless quantity for the bound states that we seek.From this equation we now determine the asymptotic behavior of the function φ(ρ)
for large values of ρ. Indeed, for ρ >> 1 (which corresponds to r >> a) we can neglectterms of order ρ−1 and ρ−2 in the eigenvalue equation, which then has the asymptoticform
d2φ
dρ2= λ2φ (ρ→∞) λ = λn,l. (4.53)
This has two independent solutions φ ∼ Ae±λρ, of which the diverging solution must berejected as being non-normalizable. From the well-behaved solution we introduce a newsubstitution, setting
u(ρ) = un,l(ρ) = φn,l(ρ)eλρ (4.54)
which makes u(ρ) a slowly varying function compared to the exponential. To determinethe function u(ρ) we evaluate
dφ
dρ= e−λρ [u0 − λu] (4.55)
d2φ
dρ2= e−λρ
£u00 − 2λu0 + λ2u
¤(4.56)
in terms of which the radial equation can be re-expressed as
u00 − 2λu0 +·2
ρ− l(l + 1)
ρ2
¸u = 0. (4.57)
The classical method of solving such an equation is to assume a power series solution
u(ρ) =∞Xj=0
cjρj = ρs
∞Xk=0
bkρk (4.58)
where in the second form we explicitly assume the first coefficient b0 is explicitly not equalto zero, and so have pulled out the leading power in the expansion. Indeed, we know thatfor small r the function φn,l(r) goes to zero as r
l+1. It follows that in the same limit thefunction u(ρ) has the leading behavior
un,l(ρ) ∼ eλρφn,l ∼ Aρl+1 +O(ρl+2). (4.59)
Thus we write
u(ρ) = ρl+1∞Xk=0
bkρk =
∞Xk=0
bkρk+l+1. (4.60)
Calculating the derivatives of this expression term-by-term and substituting into the dif-ferential equation for u(ρ) we can combine terms in powers of ρ to determine a recursionrelation for the coefficients:
k(k + l + 1)bk = 2 [(k + l)λ− 1] bk−1 (4.61)
This relation allows all coefficients to be expressed in terms of a single coefficient b0. Weobserve that
limk→∞
bk+1bk∼ 2kλk2∼ 2λk→ 0. (4.62)
Thus, the series converges for all fixed ρ. However, in general, the function u(ρ) towhich it converges will itself diverge exponentially as a function of ρ as e2λρ. This leadsto unacceptable solutions of the form φn,l ∼ eλρ. To prevent this from happening the
130 Bound States of a Central Potential
power series has to terminate, so that it reduces to a polynomial function. Thus, weconclude that acceptable solutions to the eigenvalue equation have the property that thepower series terminates: there exists some value of k ≥ 1 for which the correspondingcoefficients vanish, i.e., for which
bk+n = 0 n = 0, 1, 2, · · · . (4.63)
For this to occur, we must have
bk = 2 [(k + l)λ− 1] bk−1 = 0. (4.64)
This leads to the condition that (k + l)λ = 1, hence that the allowed values of λ satisfythe equation
λ =1
k + lk = 1, 2, · · · (4.65)
Turning this back into an equation for the energies, we find that
En,l = −λn,lE0 = −λn,lZ2ε0 (4.66)
whereλn,l =
1
k + l=1
nk = 1, 2, · · · , n = (l + 1), (l + 2), · · · . (4.67)
Thus, the spectrum of energies takes the form
En,l = −Z2ε0n2
n = (l + 1), (l + 2), · · · . (4.68)
Note that for fixed n, several l values produce the same energy. This is an accidentaldegeneracy that is not required by the rotational invariance of the problem, and reflectsother symmetries to the hydrogen atom problem that are not apparent in the presenttreatment. Expressed in the more standard way, the spectrum can be written
En = −E0n2
l = 0, 1, 2, · · · , n− 1. (4.69)
Having determined the energies, and the corresponding values of λn,l, the coefficients ofthe power series expansion are then determined by recursion. Since the series terminatesat k = n− l, the acceptable solutions take the form of polynomials
un,l(ρ) =n−l−1Xk=0
bkρk+l−1 =
nXk=l+1
ckρk (4.70)
of order n in ρ; the functions
φn,l = e−λn,lr/a0
n−1Xk=l
ck(r/a0)k (4.71)
involve polynomials of order n− 1 in r and the total wavefunctions take the form
ψn,l,m = e−r/na0
n−1Xk=l
µr
a0
¶kCkY
ml (θ,φ) (4.72)
and have energies
En = −E0n2
n = 1, 2 · · · . (4.73)
The 3-D Isotropic Oscillator 131
For each energy level En there are (2l+1)-fold degenerate angular momentum multipletscorresponding to l = 0, 1, · · · (n − 1). The degeneracy of the nth level, therefore, can becalculated as
gn =n−1Xl=0
(2l + 1) = n2. (4.74)
In spectrocopic notation the bound states |n, l,mi for angular momentum states withl = 0, 1, 2, 3, 4, 5, · · · are indicated with a letter s, p, d, f, g, h, · · · , respectively. Thus, e.g.,
ψ2pm → |2, 1,mi (4.75)
ψ3dm → |3, 2,mi (4.76)
and so on.
4.3 The 3-D Isotropic Oscillator
The Hamiltonian
H =P 2
2m+1
2mω2R2 (4.77)
=3X
α=1
·P 2α2m
+1
2mω2X2
α
¸(4.78)
is separable in Cartesian coordinates, and has eigenstates
ψn(~r) = h~r|ni = h~r|nx, ny, nzi = ψnx(x)ψny(y)ψnz(z)
that are the products of 1D oscillator wave functions
ψnx(x) =π−1/4√2nxnx!
e−q2x/2Hnx(qx)
where Hnx(q) is the Hermite polynomial, qx = βx, and β =pmω/~. The spectrum of
the isotropic oscillator Hamiltonian are the energies
En = (n+3
2)~ω n = nx + ny + nz ∈ 0, 1, 2, · · · .
which are generally degenerate due to the many different ways a positive integer n can berepresented as the sum of three other positive integers. We note for future reference thatthe product form of the eigenfunctions implies that
ψ(~r) = e−β2rfn(~r)
where fn(~r) is of polynomial order in r.Because H is clearly a scalar with respect to rotations (being a function of ~P · ~P
and ~R · ~R) we know that there also exists an basis of eigenstatesψn,l,m(~r) = h~r|n, l,mi = Fn,l(r)Yl,m(θ,φ)
common to the operators H,L2, and Lz. Indeed, we know that the function φn(r) =rFn,l(r) satisifies the equation
− ~2
2mφ00n,l +
·`(`+ 1)~2
2mr2+1
2mω2r2 − εn,l
¸φn,l = 0. (4.79)
132 Bound States of a Central Potential
As before, we simplify by multiplying through by −2m/~2 to obtain (surpressing indices)
φ00 −·`(`+ 1)
r2+ β4r2 − ν
¸φ = 0, (4.80)
where ν = 2mεn,l/~2 and β4 = m2ω2/~2. As in the Coulomb problem we first determinethe asymptotic behavior of the solution for r >> β−2, in which limit the differentialequation above simplifies to
φ00 − β4r2φ = 0, (4.81)
which has two solutoins, which at large r are dominated by the behavior exp(±β2r2/2).The decaying solution corresponds to the behavior already noted in the Cartesian form,while the growing one is clearly unacceptable. We thus introduce a new function u(r)such that
φ(r) = u(r)e−β2r2
in terms of which
φ0(r) = e−β2r2 [u0 − β2ru]
φ00 = e−β2r2£u00 − 2β2ru0 + β4(r2 − β2)u
¤.
The equation for the functoin u(r) is then found to take the form
u00 − 2β2ru0 −·β2 +
`(`+ 1)
r2− ν
¸u = 0.
We assume a power series solution consistent with the small r behavior deduced earlier,i.e.,
u(r) =∞Xk=0
bkrk+`+1 b0 6= 0.
Substitution into the differential equation for u leads, eventually, to a series of equations.The first equation of interest comes from setting the coefficients of r`+2 equal to zero andrequires for its solution that b1 = 0.For k > 1 the resulting equations reduce to a recursionrelation of the form
(k + 2)(k + 2`+ 3)bk+2 =£(2k + 2`+ 3)β2 − ν
¤bk.
From this we deduce that all coefficeints bk with k odd are identically zero. From thelarge k behavior of the even terms we deduce that if the series does not terminate it willconverge to a function that for large r behaves as exp(β2r2), which leads again to theunacceptable solution. Thus, as in the Coulomb problem, the acceptable solutions mustterminate, i.e., there exists an even integer k such that bk 6= 0, but 0 = bk+2 = bk+4 = · · · .This can only happen if, for some even integer k,
ν = 2β2 (k + `+ 3/2) k = 0, 2, 4, · · ·With the definition of ν and β this implies that
εn,l = ~ω (n+ 3/2) n = k + ` k = 0, 2, 4, · · ·Clearly, for a given value of n, the allowed ` values of orbital angular momentum satisfythe relation
εn,l = ~ω (n+ 3/2) ` = n− kThus, more than one value of ` is generally allowed for each value of n. For n even theallowed values of ` include ` = 0, 2, · · · , n, while for odd n the allowed values include ` =1, 3, · · · , n. There is, again, an accidental degeneracy in this system above that requiredby the rotational invariance of the Hamiltonian. It is a straightforward exercise to showthat the degeneracy of the nth level of the isotropic oscillator is (n+ 1)(n+ 2)/2.
Chapter 8TIME DEPENDENT PERTURBATIONS: TRANSITION THEORY
8.1 General Considerations
The methods of the last chapter have as their goal expressions for the exact energy eigen-states of a system in terms of those of a closely related system to which a constant pertur-bation has been applied. In the present chapter we consider a related problem, namely,that of determining the rate at which transitions occur between energy eigenstates of aquantum system of interest as a result of a time-dependent, usually externally applied,perturbation. Indeed, it is often the case that the only way of experimentally determiningthe structure of the energy eigenstates of a quantum mechanical system is by perturbingit in some way. We know, e.g., that if a system is in an eigenstate of the Hamiltonian,then it will remain in that state for all time. By applying perturbations, however, wecan induce transitions between di¤erent eigenstates of the unperturbed Hamiltonian. Byprobing the rate at which such transitions occur, and the energies absorbed or emittedby the system in the process, we can infer information about the states involved. Thecalculation of transition rates for such situations, and a number of others of practicalinterest are addressed in this chapter.
To begin, we consider a system described by time-independent Hamiltonian H0to which a time-dependent perturbation V (t) is applied. Thus, while the perturbation isacting, the total system Hamiltonian can be written
H(t) = H0 + V (t): (8.1)
It will be implicitly assumed unless otherwise stated in what follows that the perturbationV (t) is small compared to the unperturbed Hamiltonian H0; if we want to study theeigenstates of H0 we do not want to change those eigenstates drastically by applying astrong perturbation. In fact, we will often write the perturbation of interest in the form
V (t) = ¸V (t) (8.2)
where ¸ is a smallness parameter that we can use to tune the strength of V . We willdenote by fjnig a complete ONB of eigenstates of H0 with unperturbed energies "n; sothat, by assumption
H0jni = "njniXn
jnihnj = 1 hnjn0i = ±n;n0 : (8.3)
Our general goal is to calculate the amplitude (or probability) to …nd the system in agiven …nal state jÃf i at time t if it was known to be in some other particular state jÃiiat time t = t0: Implicit in this statement is the idea that we are going to let the systemevolve from jÃii until time t and then make a measurement of an observable A of whichjÃf i is an eigenstate (e.g., we might be measuring the operator Pf = jÃf ihÃf j). A littleless generally, if the system was initially in the unperturbed eigenstate jnii of H0 at t0;
240 Time Dependent Perturbations: Transition Theory
we wish to …nd the amplitude that it will be left in (or will be found to have made atransition to) the eigenstate jnf i at time t > t0; where now the measurement will be thatof the unperturbed Hamiltonian itself. We note in passing that if we could solve the fullSchrödinger equation
i~ ddtjÃi(t)i = H(t)jÃi(t)i (8.4)
for the initital condition jÃi(t0)i = jÃii of interest, the solution to the general problemwould be immediate. The corresponding transition amplitude would then just be theinner product Ti!f (t) = hÃf jÃi(t)i and the transition probability would be
Wi!f = jTi!f j2 =¯hÃf jÃi(t)i¯2 = hÃi(t)jÃf ihÃf jÃi(t)i: (8.5)
It is useful, in what follows, to develop our techniques for solving time-dependent problemsof this kind in terms of the evolution operator U(t; t0), or propagator, which evolves thesystem over time
jÃ(t)i = U(t; t0)jÃ(t0)i:We recall a few general features of the evolution operator
1. It is Unitary, i.e.,U+(t; t0) = U
¡1(t; t0) = U(t0; t): (8.6)
2. It obeys a simple composition rule
U(t; t0) = U(t; t0)U(t0; t0): (8.7)
3. It is smoothly connected to the identity operator
limt!t0
U(t; t0) = 1: (8.8)
4. It obeys an operator form of the Schrödinger equation
i~ ddtU(t; t0) = H(t)U(t; t0): (8.9)
5. If H(t) = H0 is independent of time, then the evolution operator takes a particularlysimple form, i.e.,
U = U0(t; t0) = e¡iH0(t¡t0)=~ : (8.10)
By comparison with what we have written above, the transition amplitude Ti!fcan be expressed as the matrix element of the evolution operator between the initial and…nal states, i.e.,
Ti!f = hÃf jÃi(t)i = hÃf jU(t; t0)jÃii; (8.11)
or, if we are interested in transitions between eigenstates of H0; we have
Tn!m = hmjU(t; t0)jni Wn!m = jhmjU(t; t0)jnij2 (8.12)
Typically, of course, it is the presence of the perturbation V (t) that renders thefull Schrödinger equation intractable. Indeed, when ¸ = 0; each eigenstate of H0 evolvesso as to acquire an oscillating phase
U0(t; t0)jni = e¡i!n(t¡t0)jni (8.13)
General Considerations 241
but no transitions between di¤erent eigenstates occur:
Wnm = ±nm:
It is the perturbation V (t) that allows the system to evolve into a mixture of unperturbedstates, an evolution that is viewed as inducing transitions between them.
Our goal, then, is to develop a general expansion for the full evolution operatorU(t; t0) in powers of the perturbation, or equivalently, in powers of the small parameter¸. To this end, it is useful to observe that the unperturbed evolution of the system isnot the goal of our calculation, involving as it does all of the unperturbed eigenenergiesof the system. Indeed, that problem is assumed to have been completely solved. It wouldbe convenient, therefore, to transform to a set of variables that evolve, in a certain sense,along with the unperturbed system, so that we can focus on the relatively slow part of theevolution induced by the weak externally applied perturbation, without worrying aboutall the rapid oscillation of the phase factors associated with the evolution occuring underH0. The idea here is similar to tranforming to a rotating coordinate system to ease thesolution of simple mechanical problems. In the present context, we expect that in thepresence of a small perturbation the unperturbed evolution changes from the form givenabove into a mixture of di¤erent states, which we can generally write in the form
U(t; t0)jni =Xm
Ám(t)e¡i"m(t¡t0)=~ jmi (8.14)
where for small enough ¸ the expansion coe¢cients Ám(t) are, it is too be hoped, slowlyvarying relative to the rapidly oscillating phase factors associated with the unperturbedevolution. As suggested above, we can formally eliminate this fast evolution generated byH0 by working in the so-called “interaction picture”.
Recall that our axioms of quantum mechanics were developed in the Schrodingerpicture in which the state of the system evolves in time
jÃSch(t)i = U(t; t0)jÃ(t0)i (8.15)
while fundamental observables of the system are associated with time-independent Her-mitian operators A = ASch. By contrast, it is possible to develop a di¤erent formulationof quantum mechanics, the so-called Heisenberg picture, in which the state of the system
jÃH(t)i = jÃ(t0)i = U+(t; t0)jÃSch(t)i (8.16)
remains …xed in time, but observables are associated with time-evolving operators
AH(t) = U+(t; t0)ASchU(t; t0): (8.17)
The kets and operators of one picture are related to those of the other through the unitarytransformation induced by the evolution operator U(t; t0) and its adjoint, and preservethe mean values, and hence predictions, of quantum mechanics in the process.
In this same spirit, it is possible to develop a formulation in which some of thetime evolution is associated with the kets of the system and some of it associated with theoperators of interest. An interaction picture of this sort can be de…ned for any system inwhich the Hamiltonian can be written in the form H = H0 + V (t); with the state vectorof this picture
jÃI(t)i = U+0 (t; t0)jÃsch(t)i (8.18)
being de…ned relative to that of the Schrödinger picture through the inverse of the unitarytransformation
U0(t; t0) = exp[¡iH0(t¡ t0)=~] (8.19)
242 Time Dependent Perturbations: Transition Theory
which governs the system in the absence of the perturbation. This form for the statevector suggests that the inverse (or adjoint) operator U+ acts on the fully-evolving statevector of the Schrödinger picture to “back out” or undo the fast evolution associated withthe unperturbed part of the Hamiltonian. In a similar fashion, the operators
AI(t) = U+0 (t; t0)ASch(t)U0(t; t0); (8.20)
of the interaction picture are related to those of the Schrodinger picture through the samecorresponding unitary transformation, but as applied to operators (we have included atime dependence in the Schrodinger operator ASch(t) on the right to take into account anyintrinisic time dependence exhibited by such operators, as occurs, e.g., with a sinusoidallyapplied perturbing …eld).
Naturally, we can de…ne an evolution operator UI(t; t0) for the interaction picturethat evolves the state vector jÃI(t)i in time, according to the relation
jÃI(t)i = UI(t; t0)jÃI(t0)i (8.21)
Using the de…nitions given above we deduce that
UI(t; t0) = U+0 (t; t0)U(t; t0) (8.22)
or, multiplying this last equation through by U0(t; t0); we obtain a result for the fullevolution operator
U(t; t0) = U0(t; t0)UI(t; t0): (8.23)
in terms of the evolution operators U0 and UI :To obtain information about transitions between the unperturbed eigenstates of
H0; then, we need the transition amplitudes
Tn!m = hmjU(t; t0)jni = hmjU0(t; t0)UI(t; t0)jni = e¡i!m(t¡t0)hmjUI(t; t0)jni (8.24)
and transition probabilities
Wn!m = jTn!mj2 = jhmjUI(t; t0)jnij2 : (8.25)
We see, therefore, that the evolution operator of the interaction picture does indeedcontain all information about transitions induced between the unperturbed eigenstates.The evolution equation obeyed by UI(t; t0) is also straightforward to obtain. By takingderivatives of U(t; t0) we establish (with t0 …xed) that
dU
dt=dU0dtUI + U0
dUIdt: (8.26)
But clearlydU
dt=¡i~[H0 + V (t)]U =
¡i~[H0 + V (t)]U0UI (8.27)
anddU0dt
=¡i~H0U0: (8.28)
From these last three equations we deduce that
i~dUIdt
= U+0 V (t)U0UI (8.29)
which we can write as
i~dUIdt
= VI(t)UI : (8.30)
General Considerations 243
Thus, the evolution operator in the interaction picture evolves under a Schrödinger equa-tion that is governed by a Hamiltonian
VI(t) = U+0 V (t)U0 (8.31)
that only includes the perturbing part of the Hamiltonian (the interaction), as representedin this picture. Since UI(t; t0) shares the limiting behavior
limt!t0
UI(t; t0) = limt!t0
U+0 (t; t0)U(t; t0) = 1 (8.32)
of any evolution operator, it obeys the integral equation that we derived earlier for evo-lution operators governed by a time-dependent Hamiltonian, i.e.,
UI(t; t0) = 1¡ i
~
Z t
t0
dt0 VI(t0)UI(t0; t0) (8.33)
and hence can be expanded in the same way in powers of the perturbation, i.e.,
UI(t; t0) =1Xk=0
U(k)I (t; t0) (8.34)
where
U(k)I (t; t0) =
µ1
i~
¶k Z t
t0
dtk ¢ ¢ ¢Z t2
t0
dt1 VI(tk)VI(tk¡1) ¢ ¢ ¢ VI(t1)
=
µ1
i~
¶k Z t
t0
dtk ¢ ¢ ¢Z t2
t0
dt1 U0(t; tk)V (tk)U0(tk; tk¡1)V (tk¡1) ¢ ¢ ¢
¢ ¢ ¢ V (t2)U0(t2; t1)V (t1) (8.35)
Combining this with (8.23) it is possible to deduce a similar expansion
U(t; t0) =1Xk=0
U(k)(t; t0) (8.36)
U(k)(t; t0) =
µ1
i~
¶k Z t
t0
dtk ¢ ¢ ¢Z t2
t0
dt1 U0(t; tk)V (tk)U0(tk; tk¡1)V (tk¡1) ¢ ¢ ¢
¢ ¢ ¢ V (t2)U0(t2; t1)V (t1)U0(t1; t0) (8.37)
for the full evolution operator. Note the structure of this is of a sum (integral) over allprocesses whereby the system evolves under H0 without perturbation from t0 to t1, atwhich time it is acted upon by the perturbation V (t1); then evolves without perturbationfrom t1 to t2; at which time it is acted upon by the perturbation V (t2); and so on. The kthorder contribution arises from all those those processes in which the system is scattered(or perturbed) exactly k times between t0 and t; with the particular times at which thoseperturbations could have acted being integrated over. It is this structure that formsthe basis for diagrammatic representations for the perturbation process, such as thoseintroduced in the context of electrodynamics by Feynman.
If the perturbation is small enough, this formal expansion for the propagator ofthe system can be trunctated after the …rst order term, and as such allows us to addressin a perturbative sense the problem originally posed. For example, if the system is at
244 Time Dependent Perturbations: Transition Theory
t = t0 initially in an eigenstate jÃ(t0)i = jni of the unperturbed Hamiltonian, the resultsof the above expansion reveal that the state of the system at time t will be given by theexpansion
jÃ(t)i =Xm
Ãm(t)jmi (8.38)
where
Ãm(t) = hmjU(t; t0)jÃ(t0)i = hmjU(t; t0)jni= hmjU0(t; t0)UI(t; 0)jni = e¡i!m(t¡t0)hmjUI(t; t0)jni: (8.39)
Truncating the expression for UI at …rst order
UI(t; t0) = 1¡ i
~
Z t
t0
dt0 VI(t0): (8.40)
and inserting the result into the expression for Ãm; keeping the lowest-order non-zeroresult for each coe¢cient, we obtain a basic equation of time-dependent perturbationtheory:
Ãm(t) = e¡i!m(t¡t0)
·±n;m ¡ i
~
Z t
t0
dt0 Vmn(t0)ei!mn(t0¡t0)
¸(8.41)
where !mn = !m¡!n is the Bohr frequency associated with the transition between levelsn and m. Clearly in this last expression, the …rst term, involving the Kronecker deltafunction is associated with the amplitude for the system to be found in the initial state,while the remaining terms give the desired (…rst-order) transition amplitudes
Tn!m = ¡ i~e¡i!m(t¡t0)
Z t
t0
dt0 Vmn(t0)ei!mn(t0¡t0) (8.42)
from which follow the corresponding transition probabilities
Wn!m = ~¡2¯Z t
t0
dt0 Vmn(t0)ei!mnt0¯2: (8.43)
For a perturbation that starts in the far distant past and dissapears in the far distantfuture, these results reduce to a particulalry simple form, in which the total transitionprobability can be written
Wn!m = ~¡2¯Z 1
¡1dt0 Vmn(t0)ei!mnt
0¯2=2¼
~2¯~Vmn(!mn)
¯2(8.44)
where~Vmn(!mn) =
1p2¼
Z 1
¡1dt0 Vmn(t0)ei!mnt
0(8.45)
is simply the Fourier transform of the perturbing matrix element connecting the twostates involved in the transition, evaluated at a frequency !mn corresponding to theenergy di¤erence between the two states involved.
As an example, we consider a 1D harmonic oscillator
H0 =P 2
2m+1
2m!2X2 (8.46)
which is initially (at t = ¡1) in its ground state when a perturbing electric …eld pulse isapplied of the form
V (t) = ¡f(t)X: (8.47)
Periodic Perturbations: Fermi’s Golden Rule 245
In this expression, f(t) = eE(t) represents the spatially uniform, but time-dependentforce exerted by the …eld on the charged harmonically-bound particle. We might, e.g.,example take a pulse envelope
f(t) = f0e¡t2=¿2 (8.48)
with a Gaussian shape that peaks at a time that for convenience we have set equal tot = 0. Our goal is to …nd the the probability that the particle is left by this pulse in thenth excited state. Provided the pulse strength is su¢ciently low, the transition probabilitycan then be written
W0!n =2¼
~2¯~Vn;0(n!)
¯2(8.49)
where~Vn;0(!) =
Xn;0p2¼
Z 1
¡1dt0 f(t0)ei!t
0(8.50)
and
Xn;0 = hnjXj0i =r
~2mw
±n;1: (8.51)
Clearly, the …rst-order transition amplitude vanishes except for the …rst excited state, i.e.,n = 1. For the Gaussian pulse, evaluation of the Fourier integral leads to the result thatlong after the pulse has passed through, the probability for the charge to be excited tothe n = 1 state is
W0!1 =f20¼¿
2
2m~!exp
¡¡!2¿2=2¢ : (8.52)
Note the transition probability becomes exponentially small as the duration of the pulse(as measured by the parameter ¿) increases, and that there is a maximum in the transitionprobability as a function of ¿ . For this perturbative result to be valid, the strength f0 ofthe …eld must be small enough that the transition probability W0!1 is small comparedto unity.
8.2 Periodic Perturbations: Fermi’s Golden Rule
An important class of problems involve perturbations that are harmonic in time, andexpressible, therefore, in the form
V (t) =£V e¡i!t + V +ei!t
¤µ(t): (8.53)
Here, µ(t) is the Heaviside step function that describes the initial application of the per-turbation at t = 0: Such a perturbation could describe, e.g., an electromagnetic waveapplied to the system at t = 0; with a wavelength much large than the system size. Weconsider here the situation in which the perturbation is simply left on and calculate, afterall the transients of the system have died down, the steady-state transition rate
¡n!m = limt!1
dWn!mdt
(8.54)
which gives the number of transitions induced per unit time by the applied perturbationbetween an initial state jni and a …nal state jmi. Using our …rst order result (8.43), thetransition probability for this situation can be written in the form
Wn!m(t) =1
~2
¯Z t
0
£Vmne
i+t + V ¤nmei¡t
¤dt
¯2(8.55)
246 Time Dependent Perturbations: Transition Theory
in which we have de…ned the quantities
+ = !m ¡ !n ¡ ! (8.56)
and¡ = !m ¡ !n + !:
Performing the integrals, we …nd
Wn!m(t) =1
~2
¯¯Vmn
¡ei+t ¡ 1¢
2i (+=2)+V ¤nm
¡ei¡t ¡ 1¢
2i (¡=2)
¯¯2
=1
~2
¯Vmne
i+t sin (+t=2)
+=2+V ¤nmei¡t sin (¡t=2)
¡=2
¯2(8.57)
which reduce to
Wn!m(t) =1
~2
½jVnmj2 sin
2 (+t=2)
(+=2)2+ jV ¤nmj2
sin2 (¡t=2)(¡=2)2
¾+
2
~2Re
½ei!mntVmnV
¤nm
sin (+t=2)
(+=2)
sin (+t=2)
(+=2)
¾: (8.58)
To put this in a form useful for exploring the long time limit, we now multiply and dividethe …rst two terms by 2¼t and the last term by ¼2 to obtain
Wn!m(t) =2¼ jVnmj2 t
~2
½1
¼
sin2 (+t=2)
2+t=2+1
¼
sin2 (¡t=2)2¡t=2
¾+2¼2 jVmnV ¤nmj
~2
·1
¼
sin (+t=2)
(+=2)
¸·1
¼
sin (¡t=2)(¡=2)
¸: (8.59)
This form is convenient, because in the long time limit, the transient oscillations in thebracketed functions tend to die away, and they approach Dirac ±-functions as T ! 1.Speci…cally, it is straightforward to establish the following representations of the Dirac±-function
±(!) = limT!1
±1(T; !) = limT!1
1
¼
sin2 (!T=2)
!2T=2(8.60)
±(!) = limT!1
±2(T;!) = limT!1
1
¼
sin (!T=2)
!=2(8.61)
by showing that the functions ±(T; !) have, as T !1; the appropriate limiting behavior(going to 1 for ! = 0; and going to 0 for ! 6= 0; respectively), and that their integralsboth approach unity for T ! 1: This allows us to write, for times t much greater thantypical evolution times of the unperturbed system
Wn!m(t) =2¼ jVnmj2 t
~2f±(+) + ±(¡)g+ 2¼
2 jVmnV ¤nmj~2
±(+)±(¡): (8.62)
Clearly, the product ±(+)±(¡) = ±(!m ¡ !n + !)±(!m ¡ !n ¡ !) vanishes, since the±-functions have di¤erent arguments. This leaves the …rst two terms, one of which mustalways vanish. If the …nal state has an energy greater than the initial, so that !m = !n+!,then the corresponding transition probability
Wn!m(t) =2¼ jVnmj2 t
~2±(!m ¡ !n ¡ !) (8.63)
Periodic Perturbations: Fermi’s Golden Rule 247
describes the resonant absorbtion of a quantum ¢" = ~! of energy; if the …nal state hasa lower energy than the initial one, so that !m = !n ¡ !; the transition probability
Wn!m(t) =2¼ jVnmj2 t
~2±(!m ¡ !n + !) (8.64)
describes the stimulated emission of a quantum ¢" = ~! of energy. The …nal form of thetransition rate for these processes can then be written
¡n!m =2¼ jVnmj2
~2±(!m ¡ !n § !) = 2¼ jVnmj2
~±("m ¡ "n § ~!): (8.65)
This is the simple form of what is referred to as Fermi’s golden rule. Since the ±-functionsmakes the transition rate formally in…nite or zero, this expression has meaning only whenthere is a distribution of …nal states having the right energy. Indeed, if we formally sumthe transition rate ¡n!m over all possible …nal states m, we can write the total transitionprobability in the form
¡n =Xm
¡n!m =Xm
2¼
~jVm;nj2 ±("m ¡ "n § ~!)
=2¼
~
Zd" ±("¡ "n § ~!)
Xm
jVmnj2 ±("¡ "m): (8.66)
If Vmn is approximately a constant over those states of the right energy to which transitionscan occur, the integral simpli…es and we end up with the second form of Fermi’s goldenrule
¡n =2¼
~jVmnj2 ½("n § ~!) = 2¼
~jVmnj2 ½("f ); (8.67)
which involves the so-called density of states
½(") =Xm
±("¡ "m) (8.68)
evaluated at the …nal energy "f = "n § ~! to which the transition can occur. Note thatthe density of states (or state distribution function) so de…ned has the property thatZ "2
"1
½(")d" = N("1; "2) (8.69)
gives the number of states of the system with energies lying between "1 and "2: Typically,situations in which Fermi’s golden rule applies are those where the …nal set of states ispart of a continuum (e.g., when a photon is given o¤ or absorbed, so that there are acontinuum of possible directions associated with the incoming or outgoing photon), andthus the density of states function ½(") is to be considered a continuous function of the…nal energy.
As an example of the application of Fermi’s golden rule, and to see how densitiesof states of the sort typically encountered are constructed, we consider a ground statehydrogen atom, with a single bound electron described by the wave function
Ã0(r) =¡¼a30
¢¡1=2e¡r=a0 ; (8.70)
to which a harmonic perturbing potential
V (~r; t) = V0 cos(~k0 ¢ ~r ¡ !t); (8.71)
248 Time Dependent Perturbations: Transition Theory
is applied, in which V0 is a constant having units of energy, and ~k0 = k0z: (The formis clearly suggestive of an electromagnetic perturbation of some sort.) Assume that theperturbation causes ionizing transitions in which the initially bound electron ends upin a “free particle state” with …nal wavevector ~k: We are interested in calculating the“di¤erential ionization rate” d¡0(µ; Á)=d for transitions to free-particle ~k-states passingthrough an in…nitesimal solid angle d centered along some particular direction (µ; Á): Toproceed, we note that the perturbation can be written in the form
V (t) = V e¡i!t + V +ei!t (8.72)
where
V =1
2V0e
i~k0¢~r: (8.73)
From Fermi’s golden rule, irreversible transitions in which a quantum ~! is absorbed(stimulated absorption) can only occur to states with …nal energies "f = "i+~! = ~!¡"0:This …nal energy is assumed to be associated with the …nal kinetic energy "f = ~2k2=2mof the ionized electron, which requires the …nal wavevector to have magnitude
k = kf =
r2m (~! ¡ "0)
~2=
r2m (~! ¡me4=2~2)
~2: (8.74)
The Fermi golden rule rate for transitions to a plane wave state of wavevector ~k havingthis magnitude can be written
¡0!~k =2¼
~
¯V~k;0
¯2±("k¡"f ) = 2¼
~
¯V~k;0
¯2±("k¡~!+"0) = 2m¼
~3k
¯V~k;0
¯2±(k¡kf ) (8.75)
where we have used the result
±
·~2
2m
¡k2 ¡ k2f
¢¸=m
~2k±(k ¡ kf ): (8.76)
Note that this last ±-function involves only the magnitude of the wavevector. The transi-tion rate d¡0(µ; Á) into all k-states passing through an in…nitesimal solid angle d along(µ; Á) is obtained by summing over all such …nal states, i.e.,
d¡0(µ; Á) =2m¼
~3k
¯V~k;0
¯2 X~k2d
±(k ¡ kf ): (8.77)
where the sum really is a symbolic way of writing an integral over all those wavevec-tors passing through the solid angle d at (µ; Á): Working in the spherical coordinaterepresentation in k-space this can be written in the formX
~k2d±(k ¡ kf ) =
Z 1
0
dk k2d ½(~k)±(k ¡ kf ) (8.78)
where ½(~k) = ½(k; µ; Á) is the density of plane wave states with wavevector ~k; i.e., thenumber of states per unit volume of k-space. To obtain this quantity, it is convenient inproblems of this sort to take the entire system to be contained in a large box of edge L;with normalized plane wave states
h~rj~ki = Á~k(~r) = L¡3=2ei~k¢r (8.79)
Periodic Perturbations: Fermi’s Golden Rule 249
that satisfy periodic boundary conditions at the edges of the box. The allowed wavevectorsin this situation are then of the form
~k =2¼
L(nx+ ny |+ nzk) (8.80)
where nx; ny; and nz are integers. The points in k-space thus form a regular cubic latticewith edge length 2¼=L; so there is exactly one state in every k-space unit cell volume of(2¼=L)3: The resulting density of states in k space
½(~k) =
µL
2¼
¶3(8.81)
is uniform, therefore, independent of ~k. Thus, the density of “ionized” states along dtakes the formX
k02d±(k0 ¡ k) =
Z 1
0
dk k2d ½(~k)±(k ¡ kf ) =µL
2¼
¶3k2d: (8.82)
Putting this into the expression given above for d¡0(µ; Á), and dividing through by d;we obtain the following expression for the “di¤erential ionization rate”
d¡0(µ; Á)
d=2m¼k
~3¯V~k;0
¯2½(~k)d =
mL3k
4¼2~3¯V~k;0
¯2(8.83)
where it is understood at this point that¯~k¯= kf as given above. This quantity gives the
number of transitions per unit time per unit solid angle along the speci…ed direction. Tocomplete the calculation we need to evaluate the matrix element
V~k;0 = h~kjV jÃ0i =V02L3=2
Zd3r e¡i~k¢~rei~k0¢~rÃ0(r)
=V02L3=2
Zd3r e¡i(~k¡~k0)¢~rÃ0(r) =
V02L3=2
~Ã0(~k ¡ ~k0) (8.84)
where, after a little hard work we …nd that
~Ã0(~q) =1p¼a30
Zd3r e¡i~q¢~re¡r=a0 =
q¼a30
8
(1 + a20q2)2
(8.85)
Combining these results we obtain, …nally:
d¡0(µ; Á)
d=
16mV 20 a20
¼~3ka0
(1 + a20j~k ¡ ~k0j2)4
=16mV 20 a
20
¼~3ka0
(1 + a20(k2 ¡ 2kk0 cos µ + k20))4
(8.86)
which is symmetric about the z-axis (independent of Á) and has a maximim along the zdirection associated with the wavevector ~k0 that characterizes the perturbation (suggest-ing the absorption of momentum from the plane wave perturbation). Note that althoughwe adopted the “box convention” for determining the density of states, corresponding fac-tors in the normalization of the …nal plane wave state led to a cancellation of any termsinvolving the size L of the box. We are free at this point to take L!1 without a¤ectingthe …nal answer.
250 Time Dependent Perturbations: Transition Theory
8.3 Perturbations that Turn On
We now consider another class of problems, one in which the measurement question thatis asked is slightly di¤erent. Consider a system subject to a time dependent perturbation
H(t) = H0 + V (t) (8.87)
in which the perturbation begins to be applied to the system at some …xed instant oftime (say t = 0), but takes a certain amount of time to develop. (The current has tobuild up in the external circuits, for example). To describe this situation, we write theperturbation in the form
V (t) = V0¸(t) (8.88)
where the function ¸(t) describes the smooth increase in the strength of the perturbationV to its …nal value V0. The function ¸(t) is unspeci…ed, but is assumed to have the generalfeatures
¸(t) =
8<: 0 for t < 0
1 for t > T(8.89)
where T is a measure of the time that it takes for the perturbation to build up to fullstrength. We note that except for the interval T > t > 0; while the Hamiltonian is ac-tually changing, the system is described by time-independent Hamiltonia: H0 initially,and H0 + V0 afterwards. During these initial and …nal intervals the evolution is readilydescribed by the corresponding eigenvectors and eigenvalues of these two di¤erent opera-tors. Borrowing from the notation we introduced previously, we denote by jn(0)i and ²(0)nthe eigenvectors and eigenvalues of H0 and by jni and "n the corresponding quantities forthe …nal Hamiltonian H = H0 + V0. Then, by assumption,
H0jn(0)i = "(0)n jn(0)i(H0 + V0) jni = "njni (8.90)
We then ask the following question. If the system is known to be in an eigenstatejn(0)i of H0 at t = 0; what is the amplitude for it to be in the eigenstate jn0i of the …nalHamiltonian H = H0 + V0 after the perturbation has fully turned on? This is clearly arelevant question, since information about the admixture of …nal eigenstates allows us topredict the subsequent evolution for t > T . So the basic question is, what happens tothe system as the perturbation is increasing to its …nal form? The general answer to thisquestion is complicated, but becomes very simple in two limiting cases: (1) a perturbationthat is applied in…nitely fast, and (2) a perturbation that is applied in…nitely slowly.
The …rst, referred to as a sudden perturbation occurs when the change in theHamiltonian occurs much more rapidly then the system (either before or after the change)can respond. In this limit, the function ¸(t) = µ(t) is essentially a Heaviside step function.The opposite limit, that in which the turn-on time T is much longer than typical evolutiontimes of the system describes what is referred to as an adiabatic perturbation.
As a useful thought-experiment that provides a mental mnemonic for remember-ing what happens in these two cases, consider what happens when a marble is placed inthe bottom (i.e., ground state) of a bowl, which is then raised slowly to some predeter-mined height. If the raised bowl is then suddenly lowered, the marble will be left hangingin air, in the “ground state” of the raised bowl, not the lowered one. It does not have time,under these circumstances to respond to the changing conditions (Hamiltonian) until longafter the bowl is in the lowered position. When, on the other hand, the bowl is loweredvery slowly, the marble stays in the “instantaneous ground state” of the bowl for eachelevation, ultimately sitting in the bottom of the bowl in the …nal lowered position. Thesefeatures also characterize the behavior of quantum mechanical systems.
Perturbations that Turn On 251
8.3.1 Sudden Perturbations
In keeping wsith the thought experiment just described, it is possible to show quite gen-erally that the state vector jÃ(t)i of a system subject to an instantaneous change in itsHamiltonian undegoes no change itself as a result of the instantaneous change in the H.In such a circumstance, the Schrödinger equation can be written (in the so-called suddenapproximation) in the formµ
i~d
dt¡H0
¶jÃ(t)i = 0 t < 0µ
i~ ddt¡H0 ¡ V0
¶jÃ(t)i = 0 t > 0 (8.91)
To understand what happens to the state vector during this change, we formally integrateacross the discontinuity in H(t) at t = 0; as follows:
djÃ(t)i = ¡i~
hH0jÃ(t)i+ µ(t)V0jÃ(t)i
idt (8.92)
Z Ã+
ádjÃ(t)i = ¡i
~
Z +"
¡"H0jÃ(t)idt¡ i
~
Z "
0
V0jÃ(t)idt (8.93)
Thus, we …nd that, for in…nitesimal "
jÃ+i ¡ jái = ¡i
~"H0jÃ+i+
i
~"H0jái ¡
i
~"V0jÃ+i (8.94)
The right hand side is proportional to "; so provided that the strength of V0 is …nite,
lim"!0 jÃ+i ¡ jái = 0: (8.95)
Hence jÃ(t)i is continuous across any …nite discontinuity in H. Thus in this limit, if thesystem is initially in an eigenstate jn(0)i of H0; it will still be in that state immediatelyafter the change in the Hamiltonian has occurred. The transition amplitude to …nd it,at that instant, in the eigenstate jn0i of H0 + V0 is just the inner product between theeigenstates of these two di¤erent Hamiltonia, i.e., :
Tn!n0 = hn0jn(0)i Wn!n0 =¯hn0jn(0)i
¯2(8.96)
As an interesting example of this class of problem, consider the beta decay of the tritiumatom, which is an isotope of hydrogen with a nucleus consisting of 2 neutrons and 1proton, so Z = 1. Suppose the single bound electron of this atom, which sees an electricpotential identical to that of hydrogen, is initially in its ground state, when the tritiumnucleus to which it is bound undergoes beta decay, a process in which the nucleus ejectsan electron with high kinetic energy (» 17 KeV), leaving behind a Helium nucleus with2 protons and a neutron. As a result of the quick ejection of the “nuclear” electron, thebound atomic electron sees the potential in which its moving change very quickly from
Vi = ¡e2
r(8.97)
to
Vf = ¡2e2
r: (8.98)
252 Time Dependent Perturbations: Transition Theory
Thus, immediately after the beta decay the electron is in the ground state of Hydrogen,
Ã1;s(~r; Z = 1) = h¹rjÃi =1p¼a30
exp (¡r=a0) (8.99)
but is moving in a potential corresponding to singly ionized Helium (He+). It is, there-fore, in a linear combination of Helium ion ground and excited eigenstates. What is theprobability amplitude that an energy measurement will …nd the electron in, say, the Ã2;sstate of the Helium ion? It is just the inner product between the Ã1s ground state ofHydrogen (with Z = 1) and the corresponding Ã2sstate (with Z = 2) for the He ion. Fora hydrogenic atom with Z = 2;
Ã2s = Ã2;0;0 =1p¼a30
µ1¡ r
a0
¶e¡r=a0 (8.100)
so the relevant transition amplitude is
T1s!2s =
Zd3r ä2s(2; r)Ã1s(1; r) =
4
a30
Z 1
0
dr r2µ1¡ r
a0
¶e¡2r=a0 = ¡1
2;
W1s!2s =1
4(8.101)
There is, therefore, a 25% chance of it ending up in this state. Such transitions can bedetected when the electron emits a photon and decays back to the ground state of the Heion. Obviously the emission spectrum for this process can be calculated by …nding thecorresponding transition probabilities for the remaining excited states of the He+ ion.
8.3.2 The Adiabatic Theorem
Perturbations that reach their full strength very slowly obey the so-called adiabatic theo-rem: if the system is initially in an eigenstate jn(0)i of H0 before the perturbation startsto change, then provided the change in H occurs slowly enough, it will adiabatically fol-low the change in the Hamiltonian, staying in an instantaneous eigenstate of H(t) whilethe change is taking place. Afterwards, therefore, it will be found in the correspondingeigenstate jni of the …nal Hamiltonian H = H0 + V0:
To see this we present a “perturbative proof” of the adiabatic theorem, by focusingon an interval of time over which the Hamiltonian changes by a very small amount. Now,by assumption, the Hamiltonian H(t) of the system is evolving very slowly in time andmay ultimately change by a great amount. Suppose, however, that there exists an instantduring this evolution when the system happens to be in an instantaneous eigenstate jniof H(t). Let us rede…ne our time scale and denote this instant of time as t = 0; and setH0 = H(0): At some time T later, the Hamiltonian will have evolved into a new operatorH(T ) = H0 + V ; where the change in H; represented by the operator V = H(T )¡H0; isassumed small, in the perturbative sense, compared to H0. We are interested in exploringhow the evolution of the system during this time interval depends upon the total timeT for this change in the Hamiltonian to take place. As already discussed, we assumethat the Hamiltonian varies in the intervening time interval T > t > 0 in such a waythat V (t) = H(t) ¡H0 = ¸(t)V ; where the function ¸(t) starts at t = 0 with the value¸(0) = 0 and increases monotonically to the …nal value ¸ = 1 when t = T: To allowfor a parameterization of the speed with which the change in H occurs, we assume thatthe function ¸ can be reexpressed in the form ¸ = ¸(t=T ) = ¸(s); with the propertiesthat ¸(0) = 0 and ¸(1) = 1: This allows us to smoothly decrease the rate at which thechange in H is being made simply by increasing the time T over which the change occurs.For convenience, we also make the assumption that ¸(s) is a monotonically increasing
Perturbations that Turn On 253
function of s = t=T for s between 0 and 1. Under these circumstances, for su¢cientlysmall perturbations V , the state at the end of this interval of time will be given to anexcellent approximation by the results of …rst order time-dependent peturbation theory:
U(T; 0)jn(0)i =Xm
Ãm(T )jm(0)i (8.102)
with
Ãn(T ) = e¡i!nT
Ãm(T ) = ¡ i~e¡i!mT
Z T
0
dt Vm;n(t)ei!mnt
= ¡iVmn~e¡i!mT
Z T
0
dt ¸(t=T )ei!mnt m 6= n: (8.103)
Performing an integration by parts, and using the limiting values of the function ¸(t=T )over this interval, leads then to the result
Ãm(t) = ¡Vmne
¡i!mT ei!mnT
!mn+Vmne
¡i!mT ei!mnT
!mn
Z T
0
dtd¸(t=T )
dtei!mnt: (8.104)
Now in the limit that the time T over which this change takes place becomes very large,the second integral becomes as small as we like. This follows from the fact that
d¸(t=T )
dt=1
T¸0(t=T ) =
d¸(s)
ds
¯s=t=T
: (8.105)
Thus the integral of interest is bounded in magnitude by the relation¯¯Z T
0
dtd¸(t=T )
dtei!mnt
¯¯ · 1
T
Z T
0
dt¯¸0(t=T )ei!mnt
¯=1
T
Z T
0
dt ¸0(t=T )
=¸(1)¡ ¸(0)
T=1
T: (8.106)
where in evaluating the last integral we have used the assumed monotonicity of ¸. Hencethe second term in the previous integration by parts is of order 1=T and becomes negligiblerelative to the …rst as T ! 1. In this limit, then, the …rst term gives for m 6= n theresult
Ãm(T ) = ¡Vmne
¡i!mT ei!mnt
~!mn= ¡Vmne
¡i!nt
"(0)m ¡ "(0)n
; (8.107)
where we have used the de…nition of !mn in terms of the corresponding eigenvalues ofH0: Thus, to this order we can write
jÃ(t)i = e¡!ntjn(0)i+ e¡i!ntXm6=n
Vmn
"(0)m ¡ "(0)n
jm(0)i
= e¡!ntjni (8.108)
where
jni = jn(0)i+Xm6=n
Vmn
"(0)m ¡ "(0)n
jm(0)i (8.109)
is the perturbative result for the exact eigenstate of H(T ) = H0 + V expressed as anexpansion in eigenstates of H0 = H(0): Thus, if the system begins the time interval in
254 Time Dependent Perturbations: Transition Theory
an eigenstate jn(0)i of H(0), it ends in an eigenstate jni of H(T ): We can now repeatthe process, presumably, by rede…ning the time such that t = T corresponds to a newtime variable t0 = 0; rede…ne H0 as H(T ) = H(t0 = 0); and proceed in the same way asabove. In this way, after many such (long) time intervals, the system has remained inthe corresponding eigenstate of the evolving Hamiltonian, which can ultimately changeby a very great amount. Provided that the change occurs su¢ciently slowly, however, thestate of the system will adiabatically “follow” the slowly-evolving Hamiltonian. Thus, theamplitude to …nd the system in an eigenstate of the …nal Hamiltonian is unity, providedit started in the corresponding eigenstate of the initial Hamiltoninan.
If the change that occurs in the Hamiltonian is not in…nitely slow, however, therewill be transitions induced to other eigenstates of H(T ). In the case of a pair of energylevels that are made to cross as a result of a time dependent perturbation it is possibleto determine the probability of transitions being induced between di¤erent correspondinglevels. The resulting analysis of such “Landau-Zener” transitions is presented in an ap-pendix. he details are a bit complicated and rely on properties of the parabolic cylinderfunctions. The end result, however, is the surpsrisingly simple expression
W = exp
µ¡ ¼V 2
~ jd"=dtj¶
(8.110)
for the transition probability between a pair of levels whose time-dependent energies crossat a rate d"=dt and which are connected by a constant matrix element V: Note that as thetime rate of change of the perturbation goes to zero, the transition probability becomesexponentially small, and can, consistent with the adiabatic theorem, be neglected provided
d"=dt << ¼V 2=~: (8.111)
8.4 Appendix: Landau-Zener Transitions
Consider a pair of energy levels connected by a constant matrix element V . If the (diag-onal) energies of the original states remain constant, then the probability amplitude tobe found in either one will oscillate in time with a frequency proportional to V and withan amplitude that depends upon the magnitude of the energy di¤erence between them.For widely separated levels very little amplitude is ever transferred from one state to theother. Even when the levels are degenerate, the transfer is complete but temporary, sincethe amplitude repeatedly oscillates entirely back to the original state. Consider, how-ever, a time-dependent perturbation that causes two widely separated levels connectedby a constant matrix element to temporarily become close, or even degenerate in energy,and then to separate. In this situation an irreversible transition can occur as a resultof the strong transfer that takes place during the limited time that the levels are nearlydegenerate, since some fraction of the amplitude will generally get “stranded” in eachstate as the levels become widely separated again in energy. Processes of this type arereferred to as Landau-Zener transitions since they were originally studied independentlyby those two authors in the context of electronic transitions in molecular systems duringcollisions. The basic idea has a wider applicability and has more recently been appliedto understand optically induced transitions between Stark-split states of atomic systemswithin the so-called dressed atom picture of Cohen-Tanoudji, et al.
To understand the essence of the Landau-Zener transition we consider two statesjÁ1i and jÁ2i; subject to a time-dependent Hamiltonian H(t) for which
H(t)jÁ1i = ~!1(t)jÁ1i+ ~vjÁ2iH(t)jÁ2i = ~!2(t)jÁ2i+ ~vjÁ1i
Appendix: Landau-Zener Transitions 255
where, for simplicity, !1(t) and !2(t) are taken to be linear functions of time such that!2(t) = ¡!1(t) = ®t=2; with ® > 0: Thus, !2 is negative for negative times and positivefor positive times, while !1 has the opposite behavior. At very large negative times thelevels are widely separated with a positive energy splitting
! = !1 ¡ !2 = ¡®tindicating that !1 > !2 for t < 0. These “bare” energy levels come together and cross att = 0, with !2 becoming larger than !1 for t > 0. The exact instantaneous eigenenergiesand eigenstates jÃ+i and jái are easily determined by diagonalizing the 2 £ 2 matrixassociated with H(t); the two roots to the secular equation
E§(t) = §~rv2 +
1
4®2t2
are indicated schematically below along with the bare energies.
-2
-1
0
1
2
-4 -2 2 4t
Clearly, at large negative times !2 corresponds to the lower branchE¡ and !1 to the upperbranch E+: The situation becomes reversed at large positive times, where !2 correspondsto E+ and !1 to E¡: Thus, up to a phase factor,
limt!1Ã+(t) = lim
t!¡1á(t) = Á2 limt!1á(t) = lim
t!¡1Ã+(t) = Á1
In the neighborhood of t = 0; the exact eigenstates are nearly equal symmetric andantisymmmetric combinations of jÁ1i and jÁ2i; and the two branches associated with theexact eigenergies exhibit the classic “avoided crossing” behavior, never coming any closertogether in energy than 2V = 2~v. Suppose that initially, as t! ¡1, the system is in theground state jÁ2i = já(¡1)i; i.e., on the lower branch E¡(t). Then, according to theadiabatic theorem, provided H(t) is varied slowly enough (®¿ 1), the system will remainon this lower branch at each instant as the system adiabatically evolves. At large positivetimes, therefore, the system will (up to a phase) be in the state jÁ1i = já(+1)i withunit probability. On the other hand, transitions between the upper and lower branchesmay occur if the variation is not su¢ciently slow.
To analyze this process, we consider the following expansion
jÃ(t)i = C1(t)e¡i©1(t)jÁ1i+C1(t)e¡i©2(t)jÁ2ifor the state of the system, where
©i(t) =
Z t
0
!i(t0)dt0 d©i=dt = !i(t):
256 Time Dependent Perturbations: Transition Theory
Substitution into the Schrödinger equation
i~ ddtjÃ(t)i = H(t)jÃ(t)i
yields the following set of …rst order di¤erential equations
idC1dt
= veiR t0! dt0C2 i
dC2dt
= ve¡iR t0! dt0C1:
for the expansion coe¢cients. We seek solutions to these equations corresponding to theboundary conditions
jC1(¡1)j = 0 jC2(¡1)j = 1;in which the system is initially in an eigenstate assocated with the lower branch E¡of the energy spectrum, and we are interested in the probability that at large positivetimes, well after the levels have separated and are no longer strongly-interacting, thesystem has made a transition from the lower branch E¡ to the upper branch E+: In thisregime E+ corresponds to the state jÁ2i: Thus, the transition probability arising from thenonadiabaticity of the perturbation is given by
P = jC2(+1)j2 = 1¡ jC1(+1)j2 :To proceed, we take another derivative and substitute back in to obtain the
following pair of second order di¤erential equations
d2C1dt2
¡ i!dC1dt
+ v2C1 = 0d2C2dt2
+ i!dC2dt
+ v2C2 = 0;
The substitutions
C1 = U1 exp
µi
2
Z t
0
! dt0¶
C2 = U2 exp
µ¡ i2
Z t
0
! dt0¶
along with the relation d!=dt = ¡® reduce these tod2U1dt2
+
µv2 ¡ i®
2+®2t2
4
¶U1 = 0
d2U2dt2
+
µv2 +
i®
2+®2t2
4
¶U2 = 0:
A …nal pair of substitutions
z = ®1=2e¡i¼=4t n = iv2=® = i°
where ° = v2=® is positive and real, put these into the standard di¤erential equations
d2U1dz2
¡µ1
4z2 ¡ n¡ 1
2
¶U1 =
d2U1dz2
¡µ1
4z2 + a1
¶U1 = 0
d2U2dz2
¡µ1
4z2 ¡ n+ 1
2
¶U2 =
d2U2dz2
¡µz2
4+ a2
¶U2 = 0
obeyed by the parabolic cylinder functions U(a; z), where here a1 = ¡n ¡ 12 and a2 =¡n+ 1
2 :The solution to the …rst of these equations having the right properties as t!§1
is the parabolic cylinder function
U1(z) = AU(¡a1;¡iz) = AU(a;¡iz);
Appendix: Landau-Zener Transitions 257
where a = n+ 12 and the constant A must be determined from the initial conditions and
the asymptotic properties of the functions U(a; x). As t ! ¡1 the argument of thefunction can be written ¡iz ! ie¡i¼=4j®1=2tj = ei¼=4R; with R ! 1 real and positive,along which path¤
U1 = AU(a;¡iz) » AU(a;Rei¼=4) » Ae¡14 iR
2
R¡n¡1e¡i¼(n+1)=4:
This clearly goes to zero as R!1 as 1=R (note that R¡n = e¡i° lnR oscillates with unitmagnitude as R increases because n is strictly imaginary). Thus this solution automati-cally satis…es the initial condition jC1 (¡1)j = jU1(¡1)j = 0: To determine the value ofA we use the other initial condition
1 = jC2(¡1)j = 1
vlim
t!¡1
¯dC1dt
¯where we have used the original di¤erential equation to express C2 in terms of the deriv-ative of C1: Now using the relation between C1 and U1 the boundary condition for U1becomes
1 = limt!¡1 v
¡1¯¡i!2U1 +
dU1dt
¯It turns out that as t!¡1 the …rst term in the brackets has precisely the same asymp-totic behavior
¡ i!2U2 = ¡i®t
2U1 » ¡1
2iA®1=2Re¡
14 iR
2
R¡n¡1e¡i¼(n+1)=4
» ¡iA2®1=2 e¡
14 iR
2
R¡ne¡i¼(n+1)=4;
as the second term
dU1dt
= A®1=2dU(a;Rei¼=4)
dR» A®1=2 d
dR
he¡
14 iR
2
R¡n¡1ei¼(n+1)=4i
» ¡ iA2®1=2e¡
14 iR
2
R¡ne¡i¼(n+1)=4:
Thus the boundary condition becomes
1 = limR!1
v¡1¯¡iA®1=2e¡ 1
4 iR2
R¡ne¡i¼(n+1)=4¯= jAj °¡1=2e¼°=4;
from which we deduce thatjAj = °1=2e¡¼°=4:
At large positive times the argument of the parabolic cylinder function can be written¡iz ! ¡ie¡i¼=4 ¯®1=2t¯ = Re¡i3¼=4. To determine the asymptotic properties in thissituation we use the identityy
p2¼U
³¡a;Re¡i¼=4
´= ¡(
1
2+ a)
ne¡i
¼2 (¡a+ 1
2)U(a;Rei¼=4) + ei¼2 (¡a+ 1
2)U(a;Re¡i3¼=4)o
¤From Abramowitz and Stegun, p.689, Eq. 19.8.1 we have for jxj >> jaj when jarg xj < ¼=2; thatU(a; x) » e¡x2=4x¡a¡1=2
yHere we use, with x = Re¡i¼=4 the expression from Abramowitz and Stegun, p. 687, Eq. 19.4.6,which gives
p2¼U (a;§x) = ¡(1
2¡ a)
ne¡i¼(
12a+
14 )U(¡a;§ix) + ei¼( 12a+ 1
4 )U(¡a;¨ix)o
258 Time Dependent Perturbations: Transition Theory
which implies that as t! +1
U1 = AU(a;Re¡i3¼=4) = A
"ei¼(n+1)U(a;Rei¼=4) +
p2¼
¡ (n+ 1)ei¼n=2U
³¡a;Re¡i¼=4
´#
» A
"ei3¼(n+1)=4e¡
14 iR
2
R¡n¡1 +p2¼
¡ (n+ 1)ei¼n=2e
14 iR
2
Rne¡i¼n=4#» A
p2¼
¡ (n+ 1)ei¼n=2e
14 iR
2
Rne¡i¼n=4
and so
jU1 (+1)j =p2¼°
¡ (n+ 1)e¡¼°=2:
The square of this gives the amplitude for the system to remain on the lower branch E¡;i.e.,
jC1 (1)j2 = limR!+1
jU1³Re¡i¼=4
´j2 = 2¼°
¡ (1 + i°) ¡ (1¡ i°)e¡¼°
= 2e¡¼° sinh¼° = 1¡ e¡¼°
and so the corresponding transition probability to the upper branch is given by theLandau-Zener formula
P = 1¡ jC1 (1)j2
= e¡¼° = exp¡¡¼V 2=~2®¢ = expµ¡ ¼V 2
~2 jd!=dtj¶:
Chapter 9SCATTERING THEORY
9.1 General Considerations
In this chapter we consider a situation of considerable experimental and theoretical inter-est, namely, the scattering of particles o¤ of a medium containing some type of scatteringcenters, such as atoms, molecules, or nuclei. The basic experimental situation of interestis indicated in the …gure below.
An incident beam of particles impinges upon a target, which maybe a cell con-taining atoms or molecules in a gas, a thin metallic foil, or a beam of particles moving atright angles to the incident beam. As a result of interactions between the particles in theinitial beam and those in the target, some of the particles in the beam are de‡ected andemerge from the target traveling along a direction (µ; Á) with respect to the original beamdirection, while some are left unscattered and emerge out the other side having undergoneno de‡ection (or undergo “forward scattering”). The number of particles de‡ected alonga given direction are then counted in a detector of some sort. The kinds of interactionsand the analyis of general scattering situations of this type can be quite complicated. Wewill focus in the following discussion on the scattering of incident particles by scatteringcenters in the target uner the following conditions:
1. The incident beam is composed of idealized spinless, structureless, point particles.
2. The interaction of the particles with the scattering centers is assumed to be elasticso that the energy of the scattered particle is …xed, the internal structure of thescatterer (if any) and, thus, the potential seen by the scattered particle does notchange during the scattering event.
3. There is no multiple scattering, so that each incident particle interacts with at mostone scattering center, a condition that can be obtained with su¢ciently thin ordilute targets.
264 Scattering Theory
4. The scattering potential V (~r1; ~r2) = V (j~r1 ¡ ~r2j) between the incident particle andthe scattering center is a central potential, so we can work in the relative coordinateand reduced mass of the system.
Under these conditions, the picture of interest reduces to that depicted below,in with an incident particle characterized by a plane wave of wavevector ~k = kz along adirection that we take parallel to the z-axis, and scattered particles emerging through ain…nitesimal solid angle d along some direction (µ; Á).
We may characterize the incident beam by its (assumed uniform) current density~Ji along the z axis. Classically
~Ji = n~vi
where n = dN=dV is the particle number density characterizing the beam. The incidentparticle current through a speci…ed surface S is then the surface integral
I =dNSdt
=
ZS
~Ji ¢ d~S:
The scattered particle current dIS into a far away detector subtending solid angle dalong (µ; Á) is found to be proportional to (i) the magnitude of the incident ‡ux densityJi, and (ii) the magnitude of the solid angle d subtended by the detector. We write
dIS =d¾(µ; Á)
dJid
where the constant of proportionality d¾(µ; Á)=d is referred to as the di¤erential crosssection for elastic scattering in the direction (µ; Á). This quantity contains all informationexperimentally available regarding the interaction between scattered particles and thescattering center. We also de…ne the total cross section ¾ = ¾tot in terms of the totalscattered current IS through a detecting sphere centered on the scattering center:
IS =
ZdIS = Ji
Zd¾(µ; Á)
dd = Ji¾tot
so that the total cross section is simply the integral over all solid angle
¾tot =
Zsphere
d¾(µ; Á)
d±:
Figure 1
General Considerations 265
of the di¤erential cross section. We note that d¾=d and ¾tot both have units of area
¾tot =IsJi
d¾
d=1
Ji
dISd
and physically represent the e¤ective cross sectional area of the target atom “seen” by theincident particle. As such it contains, in principle, information about the relative sizesof the particles involved in the collision as well as the e¤ective range of the interactionpotential V (r) without which there would be no scattering. Cross sections are oftenmeasured in “barns”, where by de…nition 1 barn = 10¡24 cm2 , which corresponds to thecross sectional area of an object with a linear extent on the order of 10¡12 cm.
Thus, given that the cross section is the primary observable of a scattering exper-iment, the main theoretical task reduces to the following: given the scattering potentialV (r); calculate the di¤erential and total scattering cross sections d¾=d and ¾tot as afunction of the energy or wavevector of the incident particle. This problem can be ad-dressed in a number of di¤erent ways. Perhaps the simplest conceptual approach wouldbe as follows:
1. Consider a particle in an initial state at t = ¡1 corresponding to a wave packet atz = ¡1 centered in momentum about ~k = kz:
2. Evolve the wavepacket accordiing to the full Schrödinger equation
i~ ddtjÃ(t)i = HjÃ(t)i
to large positive times t! +1:3. Evaluate the probability current through d along (µ; Á):
Such an approach leads to a study of the so-called S-matrix
S = limt!1U(t;¡t) = U(¡1;1):
Rather than proceed along this route, we make a few simplifying observations. First,wenote that the essential scattering process is time-independent, and can yield steady-statescattering currents, with Ji and JS independent of time. Secondly, for elastic scatteringthe particle energy is …xed and well de…ned, and it seems a shame to throw this away byforming a wavepacket of the type described. Finally, we note that the evolution of thesystem is completely governed by the positive energy solutions to the energy eigenvalueequation
(H ¡ ")jÃi = 0:This last observation leads us to ask whether or not there generally exist stationarysolutions to the energy eigenvalue equation that have asymptotic properties correspondingto the experimental situation of interest. The answer, in general, is yes and the solutions ofinterest are referred to as stationary scattering states of the associated potential V (r): Tounderstand these states it is useful to consider the 1D analogy of a free particle incidentupon a potential barrier, as indicated in the diagram. For this situation, there existsolutions in which the wave function to the left of the barrier is a linear combination of aright-going (incident) and left-going (scattered) wave, while the wave function to the rightof the barrier contains a part that corresponds to the transmitted or “forward scattered”part of the wave. We note that experimentally, the wave function in the barrier regionis inaccessible, and the only information that we can obtain is by measuring the relativemagnitudes of the forward and backward scattered waves.
266 Scattering Theory
Figure 2
Similar arguments in 3-dimensions lead us to seek solutions to the eigenvalueequation of the form
Á²(~r) = eikz + ÁS(~r)
in which the …rst part on the right clearly corresponds to the incident part of the beamand the second term corresponds to the scattered part. The subscript " indicates theenergy of the incoming and outgoing particle, which is related to the wavevector of theincoming particle through the standard relation " = ~2k2=2m: We expect that at largedistances from the scattering center, where the potential vanishes, the scattered part ofthe wave takes the form of an outward propagating wave. Hence, as r!1; we anticipatethat ÁS has the asymptotic behavior
ÁS(~r) »f(µ; Á)eikr
rr!1:
Note that this form satis…es the eigenvalue equation for large r beyond the range of thepotential, where the Hamiltonian reduces simply to the kinetic energy H ! H0 = P 2=2m:
Thus we seek solutions to the eigenvalue equation
(H0 + V ) jÁ²i = "jÁ"i
which have the asymptotic form
Á"(~r) » eikz + f(µ; Á)eikr
r
where the quantity f(µ; Á) which determines the angular distribution of the scattered partof the wave is referred to, appropriately as the scattering amplitude in the (µ; Á) directionor, alternatively, as the scattering length since it is readily determined by dimensionalanalysis that f(µ; Á) has units of length. The obvious question that arises at this point isthe following: what is the relation between the scattering length f(µ; Á) and the scatteringcross section d¾(µ; Á)=d? To anwer this question we note that, by de…nition, the currentinto the detector can be written
dIS = ¾(µ; Á)Jid:
General Considerations 267
But we can also write this in terms of the scattered current density ~JS = ~JS(r; µ; Á); i.e.,
dIs = ~JS ¢ d~S = r2JSd:Thus, classically,
Jid¾(µ; Á)
d= r2Js(r; µ; Á):
Now for quantum systems these conditions will be obeyed by the corresponding meanvalues taken with respect to the stationary state of interest, so that
d¾(µ; Á)
d= r2
hJs(r; µ; Á)ihJii :
Thus, we need operators corresponding to the current density. Classically, for a singleparticle at ~r,
~J(~r0) = n(~r0)~v = ±(~r ¡ ~r0)~v = ±(~r ¡ ~r0) ~pm:
In going to quantum mechanics we replace ~r and ~p by operators and symmeterize to ensureHermiticity. Thus,
J(~r0) =1
2m[±(~R¡ r0)~P + ~P±(~R¡ ~r0)]:
After a short calculation, the mean value of ~J(~r0) in the state jÁi is found to be
hÁj ~J(~r0)jÁi = 1
mRe
·Á¤(~r0)
~i~rÁ(~r0)
¸:
Using this, the incident ‡ux, associated with the plane wave part of the eigenstate, canbe written
hJii =¯1
mRe
·e¡ikz
~i~reikz
¸¯=~km:
By contrast, the scattered ‡ux is then given by the expression
h ~JSi = hj ~J(r; µ; Á)i = 1
mRe
·Á¤S(~r)
~i~rÁS(~r)
¸which is most conveniently expressed in spherical coordinates, for which
rr = @
@rrµ = 1
r
@
@µrÁ = 1
r sin µ
@
@Á
Using these along with the assumed asymptotic form for ÁS we …nd that
hJSir = ~km
1
r2jf(µ; Á)j2
hJSiµ = ~m
1
r3Re
·1
if¤@f
@µ
¸hJSiÁ = ~
m
1
r3 sin µRe
·1
if¤@f
@Á
¸which shows that asymptotically the angular components of current density become negli-gible compared to the radial component. Thus, from the radial component of the currentdensity we deduce that
d¾(µ; Á)
d= r2
hJs(r; µ; Á)irhJii = jf(µ; Á)j2 :
268 Scattering Theory
Thus, there is a very simple relation
d¾(µ; Á)
d= jf(µ; Á)j2
between the scattering length and the cross sections that are experimentally accessible.We now turn to the problem of actually solving the energy eigenvalue equation to …nd thestationary scattering solutions, and thereby to determine the scattering length f(µ; Á) fora given potential V (r):
9.2 An Integral Equation for the Scattering Eigenfunctions
We seek solutions to the eigenvalue equation
(H ¡ ") jÁ"i = 0H = H0 + V
in which H0 = ~2K2=2m and we assume that
V (r)! 0 as r!1 (at least as fast as r¡1)
Since we are describing a situation where we are sending in incident particles with wellde…ned kinetic energy, we expect solutions for all positive energys " ¸ 0; so we need just…nd the corresponding eigenvectors jÁ"i, which have the form
jÁ"i = jÁ0i+ jÁSi
in which jÁ0i = j~k0i is the incident part, which is an eigenstate of H0 with energy " =~2k2=2m
Á0(~r) = h~rjÁ0i = ei~k0¢~r = eikzand jÁSi is the scattered part, which should dissapear as the potential V (r) goes to zero.To proceed, we rewrite the eigenvalue equation
(H0 + V ¡ ") jÁ"i = 0in the form
("¡H0) jÁ"i = V jÁ"iand substitute in the assumed form of the solution to obtain
("¡H0) [jÁ0i+ jÁSi] = V jÁ"ior
("¡H0)jÁSi = V jÁ"i:In this last form, only the scattered part of the state appears on the left hand side. Wenote at this point that if the operator (" ¡H0) possessed an inverse, we could apply itto the left hand side to obtain a formal expression for jÁSi: The problem with this ideais that for " > 0 the operator (" ¡H0) has a large degenerate subspace with eigenvalue0; since H0 generally has a degenerate subspace for any positive energy ". Thus, strictlyspeaking, det("¡H0) = 0 and the inverse is not de…ned.
To overcome this di¢culty we employ a little analytic continuation, and de…nethe resolvent operator G(z); de…ned for all non-real z; as the operator inverse of z ¡H0;i.e.,
G(z) = (z ¡H0)¡1
An Integral Equation for the Scattering Eigenfunctions 269
since H0 has no non-real eigenvalues, this operator exists for all non-real values of thecomplex parameter z. We then de…ne the (causal) Green’s function operator G+(") asthe limit, if it exists, of G(z) as z ! "+ i´; where ´ = 0+ is a positive in…nitesimal:
G+(") = lim´!0+
("+ i´ ¡H0)¡1 ´ ("+ ¡H0)¡1 :
where"+ = "+ i´:
Thus, if the limit exists, then
lim´!0+
G("+ i´)("¡H0) = G+(")("¡H0) = 1
and we can writeG+(")("¡H0)jÁSi = G+(")jÁ"i
or, more simply,jÁSi = G+(")jÁ"i
Adding the incident part of the state jÁ0i to both sides, we then obtain the so-calledLipmann-Schwinger equation
jÁ"i = jÁ0i+G+V jÁ"iwhich is itself a representation independent form of what is often referred to as the in-tegral scattering equation. The latter follows from the Lipmann-Schwinger equation byexpressing it in the position representation. Multiplying on the left by the bra h~rj weobtain
h~rjÁ"i = h~rjÁ0i+ h~rjG+V jÁ"ior, inserting a complete set of position states,
Á"(~r) = eikz +
Zd3~r0 G+(~r; ~r0)V (~r0)Á"(~r
0):
Thus, we obtain an integral equation for the scattering eigenfunction Á"(~r) which has, wehope, the correct asymptotic behavior. To make this useful,we need to (i) evaluate thematrix elements G+(~r;~r0) = h~rjG+(")j~r0i; and (ii) actually solve the integral equation, atleast in the asymptotic regime.
9.2.1 Evaluation Of The Green’s Function
To evaluate the matrix elements of the Green’s function it is most convenient to begin thecalculation in the wavevector representation in which the operator (" ¡H0) is diagonal.Note that in k-space
h~kj("+ ¡H0)j~k0i = ("+ ¡ "k) ±(~k ¡ ~k0)where
"k = ~2k2=2m
so that
("+ ¡H0) =Zd3q j~qi ("+ ¡ "q) h~qj
and hence, as is easily veri…ed by direct multiplication,
("+ ¡H0)¡1 =Zd3q
j~qih~qj("+ ¡ "q)
270 Scattering Theory
If we set
k+ =
r2m"+~2
= k + i´ (´ = 0+)
where the positivity of ´ in this last equation follows from the positive imaginary part of"+; then we can write
G+(") =2m
~2
Zd3q
j~qih~qjk2+ ¡ q2
so that
G+(~r;~r0) =
2m
~2
Zd3q
h~rj~qih~qj~r0ik2+ ¡ q2
=2m
~2
Zd3q
(2¼)3ei~q¢(~r¡~r
0)
k2+ ¡ q2
=2m
~2
Z 2¼
0
dÁ
Z ¼
0
dµ sin µ
Z 1
0
dq q2
(2¼)3ei~qR cos µ
k2+ ¡ q2
where ~R = ~r ¡ ~r0 and R =¯~R¯: The angular integrations are readily evaluated and give
G+(~r;~r0) =
m
~2¼2R
Z 1
0
dqq sin(qR)
k2+ ¡ q2=
m
2~2¼2R
Z 1
¡1dqq sin(qR)
k2+ ¡ q2
where we have used the fact that the integrand is an even function of q. Splitting sin(qR)into exponentials and setting q0 = ¡q in the second we …nd that
G+(~r;~r0) =
m
¼~2R1
2¼i
Z 1
¡1dq
qeiqR
k+2 ¡ q2
This integral can be evaluated by contour integration in the complex q-plane using Cauchy’stheorem,which states that for a function f(z) that is analytic in and on a closed contour¡ in the complex z-plane enclosing the point z = a,
1
2¼i
I¡
f(z)dz
z ¡ a = f(a):
In our case we choose a closed path in which q runs from ¡Qto +Q and then circlesback around on a semicircle in the upper half plane, which is ultimately taken to occurat jQj =1: Since the contribution from the integrand vanishes along this latter part, theintegral over the closed contour coincides with the one of interest.
To proceed, we note that
k2+ ¡ q2 = (k+ ¡ q) (k+ + q)
which generates simple poles atq = k+ = k + i´
andq = ¡k+ = ¡k ¡ i´
only the …rst of which is enclosed by our contour. Setting
f(q) =qeiqR
k+ + q
An Integral Equation for the Scattering Eigenfunctions 271
Figure 3
we …nd that1
2¼i
I¡
f(q)dq
k+ ¡ q = ¡1
2¼i
I¡
f(q)dq
q ¡ k+ = ¡f(k+) = ¡eik+R
2
Combing this with our previous formula, and taking the limit k+ ! k we obtain theGreen’s function of interest, i.e.,
G+(~r;~r0) = ¡ meikj~r¡~r0j
2¼~2 j~r ¡ ~r0j = G+(~r ¡ ~r0):
Putting this into our integral scattering equation gives the result
Á"(~r) = eikz ¡ m
2¼~2
Zd3r0
eikj~r¡~r0jj~r ¡ ~r0j V (~r
0)Á"(~r0):
Before proceeding to solve this integral equation we should, perhaps, check to see thatit gives solutions with the correct asymptotic behavior. To this end, we note that theintegrand has contributions primarily from those regions where r0 is small, i.e., where thepotential is signi…cant. At the detector, however, the magnitude of r is very large, andV (r) is negligible. Where the integrand is signi…cant, therefore, we have j~rj >> j~r0j : Inthis limit we can write
j~r ¡ ~r0j =p(~r ¡ ~r0) ¢ (~r ¡ ~r0) =
pr2 ¡ 2~r ¢ ~r0 + r02
' rp1¡ 2~r ¢ ~r0=r2 = r ¡1¡ ~r ¢ ~r0=r2¢
' r ¡ r ¢ ~r0
where r = ~r=r is a unit vector along the direction (µ; Á) associated with the detector.Hence in this limit we can write
eikj~r¡~r0jj~r ¡ ~r0j '
eikr
re¡ikr¢~r
0
272 Scattering Theory
and so our integral equation provides a solution of the form
Á"(~r) ' eikz ¡ m
2¼~2eikr
r
Zd3r0 e¡ikr¢~r
0V (~r0)Á"(~r
0)
' eikz + f(µ; Á)eikr
r
where
f(µ; Á) = ¡ m
2¼~2
Zd3r0 e¡ikr¢~r
0V (~r0)Á"(~r
0)
is independent of r; but depends only upon r = r(µ; Á); as it should. Thus, a solution tothis equation should indeed have the correct asymptotic properties associated with thestationary scattering states of interest.
9.3 The Born Expansion
Now that we have an explicit representation for the Green’s functionG+(") we can attempta solution to the Lipmann-Schwinger equation
jÁ"i = jÁ0i+G+V jÁ"i:The traditional method of solving this kind of equation, or its integral equation equivalentis by iteration. Any approximation to jÁ"i can be substituted into the right-hand side ofthe equation to generate a new approximation . Moreover, we can formally write
jÁ"i = jÁ0i+G+V [jÁ0i+G+V jÁ"i]= jÁ0i+G+V jÁ0i+G+V G+V jÁ"i
Proceeding in this way generates the so-called Born expansion
jÁ"i = jÁ0i+G+V jÁ0i+G+V G+V jÁ0i+ ¢ ¢ ¢
=1Xk=0
(G+V )n jÁ0i:
The Born expansion gives a an expansion in powers of the potential V; and obviouslyrequires for its convergence that the e¤ect of the perturbation V on the incident wavebe small, hence jjÁS jj << jjÁ0jj; jjÁ"jj: The solution obtained by truncating the series atorder n is referred to as the nth order Born approximation to the scattered state. Wedefer till later an exploration of the approximate solutions obtained in this fashion, andinstead introduce additional ways of looking at the problem.
9.4 Scattering Amplitudes and T-Matrices
The form that we have developed for the scattering amplitude
f(µ; Á) = f(r) = ¡ m
2¼~2
Zd3r0 e¡ikr¢~r
0V (~r0)Á"(~r
0)
describes the amplitude for measuring a de‡ected particle along the direction r(µ; Á)with wavevector k: Thus, it measures a state of wavevector ~kf = kr; so we can writef(µ; Á) = f(r) = f(~kf ;~k0) in the form
f(~k;~k0) = ¡ m
2¼~2
Zd3r0 e¡ikf ¢~r
0V (~r0)Á"(~r
0)
Scattering Amplitudes and T-Matrices 273
which has the form of a projection of V jÁ"i onto the plane wave state j~kf i associated withthe wave function h~rj~kf i = ei~kf ¢~r (Note that the normalization of this plane wave state isa little di¤erent than usual, but is consistent with our choice of jÁ0i:) Thus, e.g.,
h~kf jV jÁ"i =Zd3r0 e¡ikf ¢~r
0V (~r0)Á"(~r
0)
f(~kf ;~k0) = ¡ m
2¼~2h~kf jV jÁ"i
which appears to be a matrix element of V; except that it is taken between states ofdi¤erent type. The state on the left is part of an ONB of free particle states, that onthe right is an eigenstate of the same energy in the presence of the potential, which is“smoothly connected” to the plane wave state jÁ0i = jk0i as V ! 0: It is useful tointroduce an operator T; referred to as the T matrix, or transition operator which isde…ned so that
V jÁ"i = T jÁ0i = T j~k0i:This allows us to express the scattering amplitude
f(µ; Á) = f(~k;~k0) = ¡ m
2¼~2h~kf jT j~k0i
as the matrix element of the transition operator between initial and …nal free particlestates (which are the ones that we deal with in the laboratory, outside of the targetregion). Hence the T -matrix contains all information regarding the scattering transitionsinduced by the potential. How do we evaluate T? We generate an integral equation for itfrom the Lipmann-Schwinger equation, which gives
jÁ"i = jÁ0i+G+V jÁ"i:= jÁ0i+G+T jÁ0i = (1+G+T )jÁ0i
We can compare this with the Born series to obtain
jÁ"i = jÁ0i+G+V jÁ0i+G+V G+V jÁ0i+ ¢ ¢ ¢to obtain the operator relation
G+T = jÁ"i = G+V +G+V G+V + ¢ ¢ ¢from which we deduce that
T = V + V G+V + V G+V G+V + ¢ ¢ ¢which gives a Born expansion for T in powers of the potential V . Formally we can write
T = V [1+G+V +G+V G+V + ¢ ¢ ¢ ]= V + V G+T
which is the integral equation obeyed by the T -matrix that generates the Born series.The nth order Born approximation to the T -matrix is then obtained by truncating theseries ath the nth order term. The …rst Born approximation to the T -matrix is just thescattering potential
T = T (1) = V
and so we obtain to this order
fB(~kf ;~ki) = ¡ m
2¼~2h~kf jV j~kii:
274 Scattering Theory
To evaluate this, we work in the position representation
fB(~kf ;~ki) = ¡ m
2¼~2
Zd3r h~kf j~riV (~r)h~rj~kii
= ¡ m
2¼~2
Zd3r e¡i(~kf¡~ki)¢~rV (~r):
Clearly the vector~q = ~kf ¡ ~ki
is the momentum transferred in the collision, since ~kf = ~ki + ~q. Moreover, since j~kf j =j~kij = k are the same, we can write
j~qj = 2k sin µ=2
where µ is the direction between the incoming beam and the de‡ected particle. Thus, wecan write in the …rst Born approximation
fB(~kf ;~ki) = ¡ m
2¼~2~V (~q)
where~V (~q) =
Zd3r e¡i~q¢~rV (~r)
is, up factors of 2¼; simply the Fourier transform of the scattering potential. Thus, in theBorn approximation the di¤erential scattering cross section
d¾(µ; Á)
d= jf(µ; Á)j2 = m2
4¼2~4¯~V (~q)
¯2is, up to constant factors, simply the squared modulus of the Fourier transform of thescattering potential evaluated at the wavevector ~q corresponding to the momentum trans-ferred in the scattering event.
As a special case of this formula, we can consider the case where the potential isspherically symmetric, so that V (~r) = V (r); in which case
~V (~q) = V (q) =
Z 2¼
0
dÁ
Z ¼
¡¼dµ
Z 1
0
dr r2 sin µ e¡iqr cos µV (r):
The angular integrals are readily evaluated to giveZ 2¼
0
dÁ
Z ¼
¡¼dµ sin µ e¡iqr cos µ =
4¼
qrsin(qr)
so the only remaining integral to perform is the radial one
~V (q) =4¼
q
Z 1
0
dr r sin(qr) V (r)
which will depend upon the precise form of the scattering potential.For example, if we take the so-called Yukawa (or screened-Coulomb) potential
V (r) =e2
re¡®r
Scattering Amplitudes and T-Matrices 275
then
~V (q) =4¼e2
q
Z 1
0
dr sin(qr) e¡®r
=4¼e2
®2 + q2
Thus, in this case, f(µ; Á) = f(µ) = f(q); where q = 2k sin µ=2; and
f(µ) = ¡2me2
~21
®2 + q2= ¡2me
2
~21
®2 + 4k2 sin2 µ=2
and the cross section becomes
d¾
d=
4m2e4
~4¡®2 + 4k2 sin2 µ=2
¢2 :By taking the limit that ® ! 0 we obtain the corresponding cross section, in the Bornapproximation, for the Coulomb potential
d¾
d=
m2e4
4~4k4 sin4 µ=2=
e4
16"2 sin4 µ=2:
As a second example, considering the elastic scattering of electrons by a neutralatom in its ground state with initial electron energies that are too small to excite theatom to any of its excited states. For an atom of atomic number Z; the charge density½(~r) can be written
½(~r) = e [Z±(~r)¡ n(~r)]where n(~r) is the number density of electrons in the atom at ~r and can be written
n(~r) = hÃjXi
±(~r ¡ ~ri)jÃi 'Xi
äni(~r)Ãni(~r)
where the second form holds in an independent electron approximation. The electricpotential '(~r) at a point ~r due to the charge density of the bound electrons and thenucleus satis…es Poisson’s equation
r2' = ¡4¼½(~r) = ¡4¼e [Z±(~r)¡ n(~r)]where charge neutrality implies that
Rd3r n(~r) = Z. The corresponding potential energy
seen by an incoming electron is given by
V (~r) = ¡e'(~r)so that
r2V (~r) = 4¼e2 [Z ¡ n(~r)] :We now note that if
V (~r) =
Zd3q
(2¼)3ei~q¢~r ~V (~q) ~V (~q) =
Zd3q e¡i~q¢~r ~V (~r)
then
r2V (~r) =Z
d3q
(2¼)3q2e¡i~q¢~r ~V (~q)
276 Scattering Theory
has as its Fourier transform the function ¡q2 ~V (q): We write, therefore, as the Fouriertransform of Laplaces equation
¡q2 ~V (q) = 4¼e2 [Z ¡ F (q)]where the atomic form factor
F (~q) =
Zd3r e¡i~q¢~rn(~r)
is the Fourier transform of the electronic charge density. Thus, we can solve for ~V (~q) toobtain
~V (~q) =4¼e2 [F (q)¡ Z]
q2;
which allows us to evaluate the scattering amplitude in the Born approximation
f(q) = ¡2e2m
~2[F (q)¡ Z]
q2
and the corresponding cross section
d¾
d=4e4m2
~4jF (q)¡ Zj2
q4:
Thus, measurement of the cross section for all momentum transfer allows information tobe inferred about the distribution of charge in the atom [as contained in the form factorF (q)]. For a spherically symmetric charge density it is possible, in principle, to invert thisrelation to determine the charge density ½(~r) directly.
9.5 Partial Wave Expansions
In the last section we have not really used the fact that V (r) is a spherically symmetricpotential. In this section we explore some of the simpli…cations that occur as a result ofthis fact. Speci…cally, if V = V (r); then H commutes with L2 and Lz and we know thatthere exists a basis of eigenstates common to
©H;L2; Lz
ª: Let j"; `;mi = jk; `;mi denote
such a basis for the positive energy subspace of the system of interest, where, as usual,k =
p2m"=~2:We note, in particular, that the potential V (r) = 0 is spherically symmetric, so
there must exist a basis of this sort for free particles. Let us denote by j"; `;mi(0) =jk; `;mi(0) the corresponding free particle eigenstates common to ©H0; L2; Lzª : Both setsof states satisfy orthonormality relations
hk; `;mjk0; `0;m0i = ±(k ¡ k0)±`;`0±m;m0 = (0)hk; `;mjk0; `0;m0i(0)
and have functions of the following form
Ãk;`;m(r; µ; Á) = Fk;`(r)Ym` (µ; Á) =
Ák;`(r)
rY m` (µ; Á)
Ã(0)k;`;m(r; µ; Á) = F
(0)k;` (r)Y
m` (µ; Á) =
Á(0)k;`(r)
rYm` (µ; Á)
where the functions Ák;`(r) = rFk;`(r) obey the radial equation
Á00k;` ¡µ`(`+ 1)
r2+ v(r)¡ k2
¶Ák;` = 0
Partial Wave Expansions 277
in which v(r) = 2mV (r)=~2 and k2 = 2m"=~2 ¸ 0: Note that when V = 0 this reduces to
Á(0)00k;` ¡
µ`(`+ 1)
r2¡ k2
¶Á(0)k;` = 0:
The solutions to this latter equation that are regular at the origin are well-known andrelated to the spherical Bessel functions j`(z) of order ` Speci…cally, it is found that
Á(0)k;`(r) =
r2
¼kr j`(kr);
so the free particle eigenstates of fH0; L2; Lzg are
Ã(0)k;`(~r) =
r2
¼k j`(kr) Y
m` (µ; Á):
On the other hand, provided that V (r)! 0 as r!1 faster than 1=r; then asymptoticallyboth equations (V 6= 0 and V = 0)obey the equation
Á00 + k2Á = 0 r!1;which has the general solution Á(r) » Aeikr + Be¡ikr; so that the radial dependence ofthe F (r) = Á(r)=r has the form
F (r) » Aeikr
r+B
e¡ikr
r
of a superposition of incoming and outgoing spherical waves. To conserve probability, the‡ux into the origin has to balance the ‡ux out of the origin, which imposes the requirementthat jAj = jBj ; in which case we can asymptotically write
Ák;`(r) » a` sin (kr ¡ '`) :Indeed, it can be shown from the properties of the spherical Bessel functions that the freeparticle solutions have the asymptotic behavior
Á(0)k;`(r) =
r2
¼kr j`(kr) » a` sin (kr ¡ `¼=2)
so that
'(0)` =
`¼
2
for a free particle (V = 0). When V 6= 0 it is convenient to write the phase of interest inthe form
'` = '(0)` ¡ ±` = `¼
2¡ ±`
Ák;`(r) = a` sin kr ¡ `¼=2 + ±`)where ±`is the phase shift that arises due to the potential (±` ! 0 as V ! 0). and isuniquely determined by it (whereas a` scales with the normalization of the state).
Our goal is to obtain an expansion for the stationary scattering state jÁ"i inthe complete set of states jk; `;mi and use it to obtain an expression for the scatteringamplitude f(µ; Á) expanded in spherical harmonics. In other words, we seek an expansionof the form
f(µ; Á) =X`;m
f`;mYm` (µ; Á):
278 Scattering Theory
As a preliminary simpli…cation, we note that, because of the spherical symmetry of thepotenital, there is azimuthal symmetry along the z-axis associated with the incident beam,thus, only m = 0 components exist in the expansion:
f(µ; Á) = f(µ) =X`
f`Y0` (µ) (9.1)
To proceed we express the stationary scattering states of interest as an expansion
Á"(~r) » eikz + f(µ)eikr
r=X`
A` Ãk;`;0(~r)
in states of well-de…ned angular momentum. The `th term in this expansion is referredto as the `th partial wave. In terms of the spherical harmonics, this expansion takes theform
eikz + f(µ)eikr
r=X`
A`Ák;`rY 0` (µ):
To make this useful, we now use the known expansion of the function eikz in free particlespherical waves:
eikz =X`
c` Ã(0)k;`;0(~r) =
X`
B` j`(kr) Y0` (µ):
The B` can be calculated exactly. The result is B` = i`p4¼(2`+ 1) so that
eikz =X`
i`p4¼(2`+ 1) j`(kr) Y
0` (µ):
Thus we can writeX`
·B`jl(kr) +
f`eikr
r
¸Y 0` (µ) =
XA`Ák;`(r)
rY 0` (µ):
Linear independence of the Y m` ’s implies that
B`r j`(kr) + f`eikr = A`Ák;`(r):
Now asymptotically,
B`r j`(kr) » B`ksin(kr ¡ `¼=2) = B`
2ik
heikre¡i`¼=2 ¡ e¡ikrei`¼=2
iand
A`Ák;`(r) » a` sin(kr ¡ `¼=2 + ±`)=
a`2i
heikre¡i`¼=2ei±` ¡ e¡ikrei`¼=2e¡i±`
i:
Substituting these last two equations into our previous expansion and equating coe¢cientsof eikr and e¡ikr we …nd that
a`2ie¡i`¼=2ei±` =
B`2ike¡i`¼=2 + f`
anda`2iei`¼=2e¡i±` =
B`2ikei`¼=2
Partial Wave Expansions 279
which gives two equations in the two unknown quantities a` and f`: Solving for f` we …ndthat
f` =1
k
p4¼(2`+ 1)ei±` sin ±`
so thatf(µ) =
X`
f`Y0` (µ) =
1
k
X`
p4¼(2`+ 1)ei±` sin ±`Y
0` (µ)
from which follows the expansion for the di¤erential scattering cross section
d¾
d(µ) =
1
k2
¯¯1X`=0
p4¼(2`+ 1)ei±` sin ±`Y
0` (µ)
¯¯2
The total cross section can then be written
¾tot =1
k2
Zd
¯¯1X`=0
p4¼(2`+ 1)ei±` sin ±`Y
0` (µ)
¯¯2
=1
k2
1X`=0
1X`0=0
p4¼(2`+ 1)
p4¼(2`0 + 1)ei(±`¡±`0 ) sin ±` sin ±`0
ZdY 0` (µ)Y
0`0(µ)
which reduces to
¾tot =4¼
k2
1X`=0
(2`+ 1) sin2 ±`
=1X`=0
¾`
where¾` =
4¼
k2(2`+ 1) sin2 ±` · 4¼
k2(2`+ 1)
is the scattering cross section to states with angular momentum `. Note that for freeparticles ±` ! 0 and ¾tot ! 0; as we would expect.
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