Basic Quantum Mechanics

August 4, 2011

1 Introduction

The computer industry has been driven by the urge to make devices more ecient for

handling more complex tasks and to perform the simpler tasks at a brisk pace. This has

been made possible so far by miniaturization of the integrated circuits (I.C.), so that more

circuits can be fabricated on a chip of a given size, thereby reducing the distance traversed

by an electron to perform a particular task. So far this has been in keeping with Moore's law

[give reference], which says that technology will enable the number of transistors per chip

to increase with time in a log-linear scale. However there will be a limit to this reduction

and before long one would reach a limit when there will be only one electron per device

(say a single electron transistor), and the physics governing the functioning of this device

is quantum physics and not classical physics. At microscopic scale all objects are governed

by quantum mechanics and therefore it becomes imperative to give a bit of background on

quantum mechanics before embarking on a journey of making the quantum computer.

2 State Vectors and Vector Space

2.1 Dimension and basis of a vector space

An n-dimensional quantum state can be represented by a state vector or a state function,

which is a point in n-dimensional linear vector space, dened over C, i.e. the coecients ofthe basis vectors can in general be a complex number.

A set of n vectors k is said to be linearly independent if and only if the solution of theequation,

nk=1

xkk = 0 (1)

is

x1 = x2 = ....... = xn = 0, (2)

where xk's are scalar coecents, which could in general be complex numbers. If there existsa set of scalars, not all of which are zero, such that one of the vectors in this vector space

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can be expressed as the linear combination of others,

=nk=1

xkk, (3)

then the set i is said to be linearly dependant. The basis of the vector space consistsof a set of the maximum possible numbers of linearly independent vectors belonging to that

space. These vectors 1, 2...n to be denoted by i are called basis vectors. Generally, forthe sake of convenience, in quantum mechanics, these vectors are chosen to be orthonormal

to each other, i.e., their scalar products satisfy the relation,

(i, j) = ij (4)

A basis is orthonormal if the base vectors are orthonormal to one another, and complete, if

the base vectors span the entire space. A canonical example of linear vector space that is

Euclidian three dimensional space. Here the basis vectors are the unit vectors, i, j and k ina 3 dimensional real space. In 3-D real space, they will be represented in matrix form as,

i =

100

, (5)

j =

010

, (6)and

k =

001

. (7)Any arbitrary vector in this case can be expressed as

A = Axi + Ay j + Azk, which inthis column matrix notation will be given as,

A =

AxAyAz

, (8)where, the coecients, Ax, Ay and Az are real numbers. However, for a quantum state ingeneral they will be complex numbers. Such a vector space dened over C is called a Hilbertspace. We will encounter applications of the properties of this space in the next chapter,

when we deal with qubits.

In Dirac's bra-ket notation, the state vector and is represented as |(ket) and itscomplex conjugate is represented as | (bra). Kets are elements of the Hilbert space forevery ket there is an unique bra and vice versa. The bra vectors belong to H which is thedual space of the Hilbert space H of the ket vectors. The orthonormality condition in bra-ket

notation is denoted by, |=0 (orthogonality), |=1 and | = 1 (normalization).

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2.2 Properties of this vector space

The properties of this complex n-dimensional vector space are:

(i) The states (or vectors) are commutative under addition,

(ii) Their addition is also associative,

(iii) Inner product of two states exists and the norm of a vector is positive denite,

(iv) The vector space is a complete, i.e. the basis vectors of this vector space, spans the

whole space.

(v) The basis vectors of this space constitute a linearly independent set of vectors. In

quantum mechanics we enforce a further requirement that the basis vectors have to be form

an orthonormal set, i.e. the norm is 1 and they are orthogonal to one another.

3 Observable

In quantum mechanics, any observable quantity is represented by linear operators, which

in general are represented by matrices, whose eigenvectors lie in the Hilbert space dened

above. In classical mechanics, any experimentally observable quantity can be shown to be

represented by a real-valued function on the set of all possible states of the system, which

in general is continuous. However, in quantum mechanics, this set is in general discrete, but

could be continuous in an innite dimensional Hilbert space. For example, representations

of angular momentum are in general discrete, but the representations of momentum are

continuous. A crucial dierence between observables in classical mechanics and quantum

mechanics is that in the latter case, two observables may not be simultaneously measurable.

This is mathematically expressed by non-commutativity of the corresponding operators, i.e.,

ABBA 6= O. (9)This inequality expresses a dependence of the results upon the order in which the mea-

surement of the observables are performed. Observables corresponding to non-commutative

operators are called incompatible.

In general a linear operator A has to satisfy a mathematical rule that when it acts upona state |, it transforms that state to another state | of the same vector space. Inquantum mechanics, observables are postulated to be Hermitian operators mapping Hilbert

space H onto itself. This is because, the eigenvalues of Hermitian operators/matrices arereal, signifying that observables are real and measurable quantities. A Hermitian operator

is also known as self-adjoint operator. They posses the following properties:

1. Eigenvalues of observables are real and in fact are possible outcomes of measurements

of a given observable.

2. Corresponding eigenvectors or eigenstates span the Hilbert space, which means, that each

observable generates/constitutes an orthonormal basis. After measurement over an arbitrary

state, we are left with one of the eigenstates of this operator, termed as measurement. We

will study more about measurement in the next section.

Here are some examples of observables:

1. Observables with continuous spectrum (dim(H) =): Momentum operator is repre-sented as p = i~

xand coordinate operator as x = x.

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2. Observables with discrete spectrum (dim(H) = 2) : The three operators correspondingto the the three spin coordinates of a spin- half system is represented by the three Pauli

matrices,

x =

(0 11 0

), (10)

y =

(0 ii 0

), (11)

z =

(1 00 1

). (12)

The Pauli matrices are a set of 2 x 2 complex Hermitian and unitary matrices. They satisfy

the following properties:

(i) 2i= I,(ii) det(i)=-1,(iii) Trace(i)=0,(iv) i, j = 2 i,j I(v) i, j= 2 i,j,k k,

where, i, j, k = x,y,z and I denotes an identity matrix in two dimension, [,] denotescommutation of two operators, whereas {,} denotes anti-commutation, i,j is the Kroneckerdelta and i,j,k is the Levi-Civita symbol.

4 Hamiltonian

A canonical example of an observable used extensively in quantum mechanics is the total

energy operator or the Hamiltonian, which is the sum of kinetic energy (K) and the potential

energy (V). Thus,

H = T + V (13)

The eigenvalue equation for this operator is known as the Schrdinger equation and is given

by,

i~

t| = ~

2

2m

2

x2|+ V | (14)The time independent Schrdinger equation, is given by,

H| = E|, (15)

where, E is the energy eigenvalue. Upon solving this equation, one obtains a set of eigenval-

ues and corresponding eigenvectors {En,|n}, such that the {|n}'s form an orthonormalset. These solutions are time independent and hence are called stationary states.

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5 Measurement and expectation values

In quantum mechanics and observable is a quantity that can be measured. Each dynamical

variable like spin, orbital angular momentum, can be represented by a Hermitian operator

that acts on the state of a system and whose eigenvalues correspond to the values that

dynamical variable can attain. We have seen earlier that the eigenvalues of a Hermitian

operator are real and any measurable quantity has to be real. The role of the observable

(operators) in quantum mechanics is to assign real numbers to the outcomes of a particular

measurement and these numbers are eigenvalues of that operator. Since any arbitrary state

in Hilbert space can be a superposition of the basis vectors (which in this case are the eigen-

states), an operator acting on this state will project the system into one of its eigenstates,

and this event is probabilistic. One cannot be certain beforehand which of these eigenstates

will be projected during the measurement. If the operator is the Hamiltonian, it will project

the system to one of its energy eigenstates and the system will then remain in this particular

eigenstate if the Hamiltonian is time independent. In a mixed state, we have an ensemble,

where there is a weight associated with each eigenstate, such that a measurement will project

an eigenstate with that particular weight factor. In case of a canonical ensemble, that weight

factor for the state |n is eEn . If one does a large number of measurements for an op-erator, one will obtain an average value for that operator, also known as the expectation

value. Thus if we have an arbitrary state, | = nk=1 akk, then the expectation value ofan operator A, is given by,

A = |A||| (16)

6 Postulates of quantum mechanics

We will review the fundamental postulates of quantum mechanics in this section. The

postulates are as follows:

(i) The state of physical systems are represented by vectors in Hilbert space.

(ii) Any measurable or observable quantity is represented by Hermitian operators.

(iii) When a measurement is performed, the state is projected onto one of the eigenstates of

the operator. This is also known as the collapse of an arbitrary wave-function onto one of

its eigenstates.

(iv) The average value of such a measurement is given by the expectation value of that

operator.

7 Time evolution of expectation values

So far we have been dealing with only time independent Hamiltonians. In such a case, we

have to solve the time dependent Schrdinger equation. Quantum mechanics enables us

to calculate the time evolution of a dynamical system provided the Hamiltonian is dened

and the initial state is suitably specied. The Hamiltonian must include all the interactions

that the system is subjected to and here we will deal with closed systems, where eect of

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external elds or forces are neglected. In the so-called Schrdinger picture of quantum

mechanics, the dynamics is governed by the time evolution of the state, which results from

the solution of the equation,

i~d

dt|(t) = H |(t) (17)where | (t) denotes the state of the system at any one time t, H is a densely-dened self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and ~ is the reducedPlanck constant. H is the total energy of the system.

Alternatively, by Stone's theorem one can state that there is a strongly continuous one-

parameter unitary group U(t) : H?H such that

|(t+ s) = U(t) |(s) (18)for all times s, t. There exists a self-adjoint Hamiltonian H such that

U(t) = e(i/~)tH (19)

is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that H

does not depend on time and that the perturbation starts at t0 = 0; otherwise one must usethe Dyson series, formally written as

U(t) = T {exp(i/~)t

t0

dtH(t)} , (20)

where T , is Dyson's time-ordering symbol. This symbol permutes a product of non-commutingoperators of the form B1(t1) B2(t2) Bn(tn) into the uniquely determined re-orderedexpressionBi1(ti1) Bi2(ti2) Bin(tin) with ti1 ti2 tin . The result is a causalchain, the primary cause in the past on the utmost r.h.s., and nally the present eect on

the utmost l.h.s..

An alternative picture of quantum mechanics, known as the Heisenberg picture, focuses

on observables and instead of considering states as varying in time, it regards the states

as xed and the observables as changing. To go from the Schrdinger to the Heisenberg

picture one needs to dene time-independent states and time-dependent operators. Thus,

| = |(0)A(t) = U(t)AU(t). (21)It can easily veried that the expected values of all observables are the same in both pictures,

| A(t) | = (t) | A | (t) (22)and that the time-dependent Heisenberg operators satisfy

i~d

dtA(t) = [A(t), H]. (23)

It is assumed that A is not time dependent in the Schrdinger picture.

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The third picture, which is the so-called Dirac picture or interaction picture has time-

dependent states and observables, evolving with respect to dierent Hamiltonians. This

picture is most useful when the evolution of the observables can be solved exactly, conn-

ing any complications to the evolution of the states. For this reason, the Hamiltonian for

the observables is called "free Hamiltonian" and the Hamiltonian for the states is called

"interaction Hamiltonian". The time evolution equation is given by,

i~d

dt|(t) = Hint(t) |(t) i~ d

dtA(t) = [A(t), H0]. (24)

The Heisenberg picture is the closest to classical Hamiltonian mechanics, for example, the

commutators appearing in the above equations directly translate into the classical Poisson

brackets; and the Schrdinger picture is considered easiest to visualize and understand by

most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture

is the one used in perturbation theory, and is specially associated to quantum eld theory

and many-body physics.

Similar equations can be written for any one-parameter unitary group of symmetries of

the physical system. Time would be replaced by a suitable coordinate parameterizing the

unitary group, for instance, a rotation angle, or a translation distance and the Hamilto-

nian would be replaced by the conserved quantity associated to the symmetry, for example,

angular or linear momentum.

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