3 Semiclassical and quantum theories of ra-

diation

Quantum field theory, whose formulation is the main goal of this book, grewout in the second quarter of the XX century mainly from the attempts atformulating quantitative theories of two classes of observed phenomena. Oneclass was related to the emission and absorption of light by atoms, moleculesand (in the later period) nuclei that is, processes which are all due to theinteraction of charged matter particles with the electromagnetic field. Theother class of phenomena formed β decays of nuclei whose early and success-ful theory was given by E. Fermi in 1932. The first class of problems ledinstead to the formulation of Quantum Electrodynamics (QED) as the firstphysically satisfactory relativistic quantum field theory. However, even thesimplest systematic formulation of QED, by quantizing the electromagneticfield coupled to other degrees of freedom (matter particles or other fields),is rather cumbersome (see section 11.7). Therefore we begin by presentingthe early theory of the interaction of matter particles (electrons) describedby the nonrelativistic quantum mechanics (based on the Schrodinger waveequation) with radiation described classically. We will derive the main for-mulae allowing to compute rates (probabilities per unit time) of induced andspontaneous transitions which (after refinements related to accounting forthe electron spin) allowed to explain the bulk of the spectroscopic data. Thesemiclassical approach accounts rather naturally for absorption and stimu-lated emission processes (in this part it essentially the application of the timedependent perturbative expansion of section 2.5) but the traetement withinit of the spontaneous emission relies on rather hand-waving arguments. Alsothe influence of the emission and absorption on the radiation itself is ignored.To lend some support to the results obtained in this approach we will, there-fore, compare these predictions with the Einstein’s analysis of the emissionand absorption processes which is based on general statistical arguments.Finally, ignoring the complications (which will be dealt with only in section11.7) we will quantize the free radiation field. Introducing then the couplingto the matter particle(s) we will obtain a quantum version of the radiationtheory. This theory, while still not being complete, allows to compute willnot only allow to rederive the results obtained in the semiclassical approachbut also treat processes in which light is scattered on matter.

3.1 Gauge invariance of electrodynamics

The semiclassical radiation theory, as it will be presented here, consists oftwo elements: the classical Maxwell equations satisfied by the electric E

54

and magnetic B fields and the ordinary quantum mechanics (based on theSchrodinger Equation) of a single charged matter particle interacting with theelectromagnetic field. The second ingredient - the single particle Schrodingerquantum mechanics - can, of course, be modified to take into account spinor be replaced by the nonrelativistic quantum mechanics of many particles(e.g. in the second-quantization formulation developed in section 5) takinginto account also static interactions between charged matter particles).

The Maxwell equations in the (illegal but in theoretical physics mostnatural) Gauss units read

∇×E+1

c

∂B

∂t= 0 , ∇·B = 0 , (3.1)

∇×B− 1

c

∂E

∂t=

4π

cJ , ∇·E = 4πρ . (3.2)

The current density J and the charge density ρ cannot be independent: tak-ing the time derivative of both sides of the Gauss law (the last of the fourequations) and expressing the time derivative of E using the third equationone learns that J and ρ must be related by the continuity equation

∂ρ

∂t+∇·J = 0 , (3.3)

which is, therefore, the necessary consistency condition for the joint systemof equations (3.1) and (3.2).

The equations (3.1) are automatically satisfied if instead of the two vectorfields E and B one works with the scalar and vector potentials ϕ and A

(which together form a relativistic four-vector Aµ), such that

B = ∇×A , E = −∇ϕ− 1

c

∂A

∂t. (3.4)

The potentials ϕ and A satisfy the equations which follow from inserting E

and B given by (3.4) into the second pair of Maxwell equations (3.2):

∇×∇×A+1

c2∂2A

∂t2+

1

c∇∂ϕ

∂t=

4π

cJ ,

−1

c

∂

∂t∇·A−∇2ϕ = 4πρ . (3.5)

Since in the Cartesian coordinates ∇×∇×A = ∇(∇ ·A)−∇2A the above

equations can be rewritten in the form

1

c2∂2A

∂t2−∇

2A+∇

(

1

c

∂ϕ

∂t+∇·A

)

=4π

cJ , (3.6)

1

c2∂2ϕ

∂t2−∇

2ϕ− 1

c

∂

∂t

(

1

c

∂ϕ

∂t+∇·A

)

= 4πρ . (3.7)

55

Time evolution of the state-vector representing the electron of mass Mand the electric charge qe (the electron charge q in units of e > 0 is q = −1)bound in an atom and interacting with the electric and magnetic fields is inturn governed by the Schrodinger equation:1

i~∂

∂tψ =

[

1

2M

(

−i~∇− qe

cA)2

+ qeϕ+ V

]

ψ (3.8)

=

[

− 1

2M∇

2 +iqe~

McA·∇+

iqe~

2Mc(∇·A) +

q2e2

2Mc2A2 + qeϕ+ V

]

ψ ,

in which V (r) represents the potential binding the electron in the atom. V (r)is also of the electromagnetic origin but is treated here as independent of theradiation field represented by ϕ and A.

It is known that the potentials ϕ and A are not uniquely determined bythe fields E and B: the potentials

A′ = A−∇θ , ϕ′ = ϕ+1

c

∂θ

∂t, (3.9)

constructed using an arbitrary function θ(t, r) give, via (3.4), the same fieldsE and B as the original ϕ and A. The transformation A → A′, ϕ → ϕ′ iscalled gauge transformation. The corresponding gauge transformation of thewave function ψ is:

ψ → ψ′ = exp (−iqeθ/~c)ψ, (3.10)

that is, if ψ(t, r) satisfies the Schrodinger equation (3.8) with the originalpotentials ϕ and A, then the gauge-transformed function ψ′(t, r) satisfiesit with ϕ′ and A′ given by (3.9). This becomes obvious if the Schrodingerequation (3.8) is rewritten in the equivalent form

(

i~∂

∂t− qeϕ

)

ψ =1

2M

(

−i~∇− qe

cA)(

−i~∇− qe

cA)

ψ + V ψ ,

because, as it is easy to check,

(. . .A′ . . .)ψ′ = e−iqeθ/~c(. . .A . . .)ψ ,

(. . . ϕ′ . . .)ψ′ = e−iqeθ/~c(. . . ϕ . . .)ψ .

Using this freedom one can, for example, work in the so-called Lorentz gaugein which the potentials satisfy the condition2

1

c

∂ϕ

∂t+∇·A = 0 . (3.11)

1If the spin degree of freedom becomes important the Schrodinger equation (3.8) shouldbe replaced by the Pauli equation.

2This does not fix the gauge completely: it is still possible to change ϕ and A asin (3.9), provided the gauge function θ satisfies the equation ∂2θ = 0, where ∂2 is thed’Alembertian ∂2 ≡ (1/c2)∂2/∂t2 −∇

2.

56

In section 11.7 it will be seen, however, that the canonical quantization ofthe system consisting of the electromagnetic field coupled to particles (or theelectromagnetic field coupled to other fields) is most easly performed in theradiation or Coulomb gauge in which

∇·A = 0 . (3.12)

In this gauge, which will be adopted also here, effects due to the radiation andeffects due to the static Coulomb interactions of charges are separated. Sincein the semiclassical treatement of this section we will mostly consider theelectromagnetic field propagating freely (recall, we neglect the back reactionof matter atoms on the radiation!), that is with ρ = 0, it will be possible3 tochoose the gauge so that ∇·A = 0 and ϕ = 0.

3.2 Induced transition

In the gauge (3.12) the Maxwell equations in the empty space reduce to thesimple single wave equation

1

c2∂2A

∂t2−∇

2A = 0 . (3.13)

Its general solution can be obtained as a superposition of the plane waves

A(t, r) = A0 ǫ(k) e−i(ωt−k·r) + c.c., (3.14)

where ǫ(k) ≡ ǫ(n) (n = k/|k|) is a complex polarization vector satisfyingthe condition ǫ

∗ · ǫ = 1 and A0 is a complex amplitude. The equation (3.13)

3Given ϕ(t, r) and A(t, r) satisfying field equations (3.6), (3.7) construct θ(t, r) as

θ(t, r) = −∫ t

t0

dt′ c ϕ(t′, r) + θ(r) ,

with still unspecified θ(r). Then ϕ′(t, r) = ϕ(t, r) + (1/c)∂θ(t, r)/∂t = 0 and

∇·A′ = ∇·A+

∫ t

t0

dt′ c∇·(∇ϕ(t′, r)) −∇2θ(r) .

As ϕ has been assumed to satisfy the equation (3.7) (i.e. (3.5)), the integral is

∫ t

t0

dt′ c

(

−4πρ(t′, r)− 1

c

∂

∂t′∇·A(t′, r)

)

= −∇·A(t, r) +∇·A(t0, r)− 4πc

∫ t

t0

dt′ ρ(t′, r) .

Hence, if ρ = 0 it is sufficient to take θ(r) such that ∇2θ(r) = ∇·A(t0, r).

57

is satisfied provided ω = c|k| and the radiation gauge ∇·A = 0 imposes thetransversality condition

k·ǫ(k) = 0 . (3.15)

Thus, for a given direction of the wave vector k there are only two indepen-dent polarization vectors ǫ(k, λ), λ = ±1 such that ǫ∗(k, λ′) · ǫ(k, λ) = δλ′λ.

The electric and magnetic fields are then given by

E = −1

c

∂A

∂t= iA0 |k| ǫ(k) e−i(ωt−k·r) + c.c. (3.16)

B = ∇×A = iA0 k×ǫ(k) e−i(ωt−k·r) + c.c. (3.17)

The Poynting vector P (which is the 0i component of the energy-momentumtensor of the electromagnetic field) which is related4 to the energy densityw = (E2 +B2)/8π by P = n cw giving the flux of the energy carried by theelectromagnetic field then is

P =c

4πE×B (3.18)

= −c|k|4π

[

A0ǫ e−i(...) −A∗

0ǫ∗e+i(...)

]

×[

A0k×ǫ e−i(...) −A∗0k×ǫ

∗e+i(...)]

.

In the case of a monochromatic wave one is usually interested in the Poyntingvector averaged over the period T = 2π/ω:

〈P〉T =c

4π|A0|2 |k| [ǫ× (k× ǫ

∗) + c.c.] =ω

2π|A0|2 k = n

ω2

2πc|A0|2 , (3.19)

where we have used ǫ∗ · ǫ = 1 and (3.15). The intensity of the radiation, i.e.

the energy falling per unit time on a unit area perpendicular to k, equalstherefore |〈P〉T | = ω2|A0|2/2πc.

We can now apply the perturbative approach developed in section 2 andin particular in section 2.5 to the time evolution of the quantum state ofan atom interacting with the electromagnetic radiation. We are interestedin computing probabilities of transitions (induced by the radiation) betweenthe stationary states of the Hamiltonian

H0 = − ~2

2M∇

2 + V (r) , (3.20)

of the single electron. If the whole atom has only a single electron (aHydrogen-like atom), V (r) = −Ze2/r. Our treatement of many-electron

4The energy falling onto the area |∆s| (perpendicular to P) in the time interval ∆tequals (P ·∆s)∆t. It should be equal to the energy which occupies the cylinder of heightc ·∆t and the base |∆s|. This shows that the energy density w is just |P|/c.

58

atoms will be instead only approximate: it assumes that each electron ofsuch an atom can be considered as moving in a spherically symmetric po-tential V (r) representing the attractive force of the nucleus and the mean(repulsive) force of all the remaining electrons. (All the simplifications madecan be, of course, relaxed). We assume that the electron on which we con-centrate is initially in one of the stationary bound states of the Hamiltonian(3.20)

The perturbation Vint simplifies in the adopted radiation gauge to5 (recallthat q = −1)

Vint =iqe~

McA·∇ ≡ − qe

McA·p =

e

McA(t, r)·p . (3.21)

We have dropped the term (q2e2/2Mc2)A2 present in the Schrodinger equa-tion (3.8) because its effects are in most cases strongly suppressed comparedwith the ones of the term retained in (3.21).

If one is interested in transitions to stationary states of the continuouspart of the spectrum ofH0 (3.20), i.e. in the probability of (partial) ionizationof the atom by the monochromatic electromagnetic wave, the formalism ofsection 2.5 can be applied directly with A(t, r) in the form of the plane wave(3.14), whose time dependence is harmonic (with the frequency ω). Theprobability that the electron will make in a unit time the transition from adiscrete atomic state |i〉 to the group of states characterized by the electronmomentum ~p of length ~|p| =

√

2M(Ei + ~ω) and the direction in theelement dΩp of the solid angle is then given by (2.42) with

〈p−|O|i〉 = − ie~

Mc

∫

d3ru(−)∗p (r)A0 e

ik·rǫ(k)·∇ui(r) (3.22)

where u(−)p (r) = 〈r|p−〉 (cf. the discussion in section 2.4), normalized so

that 〈p′−|p−〉 = (2π)3δ(3)(p′ − p), is the Coulombic (out) wave function cor-

responding to the continuous H0 eigenvalue ~2p2/2M . The (differential)

density of states in (2.42) is, with this normalization, the same as in (2.43).Using this formalism the photoelectric effect cross section can be calculated.

Such a simple approach cannot be directly applied to transitions betweenthe states |i〉 and |f〉 belonging to the discrete part of the spectrum of H0

(3.20). Exactly monochromatic electromagnetic wave, if tuned exactly to theenergy difference6 Ef −Ei, would cause, as in the example treated in section

5In the third form of Vint it is made explicit (by using hats) that from the point of viewof the quantum mechanics of the electron, the space argument r of the classical externalpotential A becomes the Hilbert space operator.

6We disregard here the fact that true excited atomic energy levels have always nonzero

59

2.5 in the two-state approximation, continuous transitions, forth and backbetween the states |i〉 and |f〉 and the probabilities of finding the electron inany of these states could not be computed perturbatively.

However, atoms usually do not interact with ideal monochromatic elec-tromagnetic waves (unless these are artificially produced) and we would liketo consider here transitions between the discrete states |i〉 and |f〉 occuring inan atom immersed in the radiation produced by other atoms (of the same gasor coming from another thermal light source). In particular, we would liketo express probabilities of atomic transitions between |i〉 and |f〉 induced bysuch a radiation in terms of a quantity characterizing the energy containedin a radiation flux which falls on the atom (from a fixed direction). To thisend we will first construct a simple model of the electromagnetic radiation towhich the atom can be exposed. With this aim we consider a little bit morerealistic superposition of plane waves (3.14), a wave train, e.g. of the form

A(t, r) =

∫ ∞

0

dω[

ǫ(k)A(ω) e−i(ωt−k·r) + c.c.]

, (3.23)

with a (complex) profile A(ω). The Poynting vector (3.18) integrated overtime then is∫ +∞

−∞

dtP =c

4π

∫ ∞

0

dω

∫ ∞

0

dω′

∫ +∞

−∞

dt (3.24)

[

A(ω)A∗(ω′) |k| ǫ×(k′×ǫ′∗) e−i(ω−ω′)tei(k−k′)·r + . . .

]

,

where the ellipses in the square bracket stand for other three terms whichhave exponential factors e+i(ω−ω′)t, e+i(ω+ω′)t and e−i(ω+ω′)t, respectively andcontractions ǫ

∗ · ǫ′ (the second one) or ǫ∗ · ǫ′∗, ǫ · ǫ′ (the last two). The

integration over time in (3.24) produces the delta functions in the standardway

∫ +∞

−∞

dt e−i(ω−ω′)t = 2πδ(ω − ω′) ,

etc. The two terms with e±i(ω+ω′)t in the bracket in (3.24) give then zerobecause the integrals over frequencies are taken only from 0 to ∞ (the fre-quencies are by definition positive whereas the delta functions resulting from

widths; owing to this fact tuning the almost monochromatic wave becomes possible andthe approach of section 2.5 in principle could be applied but computing the density ρ(E) ofthe excited level is practically impossible. From another perspective, the nonzero widths ofexcited energy levels result from spontaneous emission processes (which will be discussed,using the semiclassical approach, in section 3.3) which formally are effects of the first orderperturbation (3.21). Doing perturbative calculations rigorously in the first order one has,therefore, to treat the levels as truly discrete ones (having zero width). This is yet anotherillustration of the perturbative unitarization.

60

the integration over dt enforce ω = −ω′). From the remaining two terms oneobtains

∫ +∞

−∞

dtP = n

∫ ∞

0

dωω2

c|A(ω)|2 , (3.25)

which represents the total energy which passed through the unit area perpen-dicular to the wave vector k. Although formally the wave packet (3.23) hasneither the beginning nor the end, in realistic situations, due to the singleGaussian-like shape of the profile A(ω), there is a well defined period of finitelength during which almost the whole energy given by (3.25) is absorbed. Wewill assume that this period ∆τ , which is determined by the inverse of thespread ∆ω of the profile A(ω) around its characteristic frequency, is reason-ably short.

If the considered atom is irradiated for a longer time, the radiation isusually an incoherent superposition of waves of the form (3.23) emitted bydifferent sources at different times (the case of an atom placed in the laserlight is different!). This is for example so, if the radiation originates fromother atoms making spontanous transitions (to be considered later) at dif-ferent moments. Such light is said to consist of incoherent packets (wavetrains). In order to construct a simple model of such an electromagneticradiation we first notice that the frequency profiles A(ω) and A′(ω) of twopackets (3.23)A(t, r) andA′(t, r) = A(t−∆t, r), that is, which are separatedin time by ∆t, but are otherwise identical, are related by A′(ω) = A(ω)eiω∆t.We will assume for a moment - to simplify things as much as possible -that the radiation is composed of packets like (3.23) of two different types|A1(ω)| 6= |A2(ω)| and that packets of electromagnetic waves of each typefall on the atom from a fixed direction for a longer time, separated by thetime intervals ∆t1 and ∆t2, respectively. In addition we assume that theprofiles of the consecutive packets differ, in addition to the factors eiω∆ti , byadditional phase factors which are completely random. In other words, weassume that the the total frequency profile of the electromagnetic radiationfalling on the atom has the form

A(ω) = A1(ω)(

eiα(0)1 + eiα

(1)1 eiω∆t1 + . . .+ eiα

(p1)1 ep1iω∆t1

)

+A2(ω)(

eiα(0)2 + eiα

(1)2 eiω∆t2 + . . .+ eiα

(p2)2 ep2iω∆t2

)

,

in which pℓ are the numbers of superposed packets of type ℓ. As in (3.25),the Poynting vector corresponding to such a radiation integrated over time isgiven by |A(ω)|2 times n ω2/c integrated over dω. We are now interested in

what happens when more and more wave packets with random phases α(m)1

and α(m)2 are superposed. In the limit p1 → ∞, p2 → ∞ with the interval

61

p1∆t1 ≈ p2∆t2 = ∆t held fixed one gets

|A(ω)|2 = |A1(ω)|2p1∑

m,m′=0

eiω(m−m′)∆t1ei(α(m)1 −α

(m′)1 )

+ |A2(ω)|2p2∑

m,m′=0

eiω(m−m′)∆t2ei(α(m)2 −α

(m′)2 )

+ 2ReA1(ω)A∗2(ω)

p1∑

m=0

p2∑

m′=0

eiω(m∆t1−m′∆t2)ei(α(m)1 −α

(m′)2 )

→ p1|A1(ω)|2 + p2|A2(ω)|2 ,

because all the interference terms (between packets of the same and differenttypes) tend to cancel owing to the complete randomness of the phases. As aresult, the energy falling on the atom over a period sufficiently long comparedto ∆t1 and ∆t2 is the same as if the wave trains did not interfere at all. This iswhat it means that the incoming radiation consists of incoherent wave trains(packets). Moreover if one assumes that that ∆tℓ >

∼ ∆τℓ, the the time interval∆t ≈ p1∆t1 ≈ p2∆t2 is approximately equal to the period over which all theenergy carried by the radiation passes through the unit area perpendicularto the direction of k. Therefore, the mean energy falling per unit time on aplane of unit area perpendicular to the vector k can be computed as

lim∆t→∞

1

∆t

∫ ∞

0

dωω2

c|A(ω)|2

=

∫ ∞

0

dωω2

c

(

1

p1∆t1p1|A1(ω)|2 +

1

p2∆t2p2|A2(ω)|2

)

= r1

∫ ∞

0

dωω2

c|A1(ω)|2 + r2

∫ ∞

0

dωω2

c|A2(ω)|2 ,

where rℓ ≡ 1/∆tℓ is the number of wave trains of the type ℓ arriving from theconsidered direction (set by k) per unit time. This reasoning can be easilygeneralized to superpositions of more types of packets. In general, therefore,the energy of the radiation flux falling on the atom from the direction k canbe characterized by its so-called radiation flux spectral density I(ω) which inour simple model of the radiation flux can be defined as

I(ω) =ω2

c

∑

ℓ

rℓ |Aℓ(ω)|2 . (3.26)

I(ω)dω represents the mean energy of the radiation falling per unit time onthe plane (perpendicular to k) of unit area in the frequency interval (ω, ω+dω). If the system consisting of atoms and radiation is in thermal equilibrium,

62

the spectral density I(ω), or better, the energy density ρ(ω) = I(ω)/c, of theradiation in which each individual atom is immersed is constant.

Having a model of the radiation flux, we can now compute probabili-ties per unit time of various transitions (occuring in an ensemble of atomsprepared in the same state - see the discussion in section 2.5) between thediscrete eigenstates of H0 (3.20). To this end we first go back to the for-mula (2.8) and, assuming that the atom was prepared in the discrete state|i〉 in the far past (t = −∞), compute, using (3.21) as the perturbation withA(t, r) being a single Gaussian-like wave train (3.23), the total probabilityof the atom’s transition to another discrete state |f〉 6= |i〉 induced by thewave train (3.23). In the first order approximation the relevant coefficient

afi = a(1)fi + . . . in the expansion of the electron’s state-vector |Ψ(∞)〉I is

a(1)fi = − 1

~c

∫ ∞

−∞

dt

∫ ∞

0

dω

[

ωA(ω) ei(ωfi−ω)t ie

Mω〈f | eik·r p |i〉·ǫ(k)

+ωA∗(ω) ei(ωfi+ω)t ie

Mω〈f | e−ik·r p |i〉·ǫ∗(k)

]

.

Therefore, in this approximation, the total transition probability is (again,only one of the two delta functions resulting from the integration over timecontributes, depending of whether ωfi is positive or negative)

∣

∣

∣a(1)fi [A]

∣

∣

∣

2

=4π2

~2c2|ωfiA(ωfi)|2 |ǫ(k)·p(+)

fi |2 if Ef > Ei , (3.27)

∣

∣

∣a(1)fi [A]

∣

∣

∣

2

=4π2

~2c2|ωifA(ωif)|2 |ǫ∗(k)·p(−)

fi |2 if Ef < Ei , (3.28)

where

p(±)fi ≡ ± ie

Mω〈f | e±ik·r p |i〉

∣

∣

∣

∣

ω=|ωfi|

. (3.29)

In the formulae (3.27), (3.28) we have explicitly indicated the functional

dependence of the coefficients a(1)fi on the profile A(ω).

Although (3.27) and (3.28) are formally the transition probabilities cor-responding to t = ∞ (i.e. obtained after waiting infinitely long), in realisticcases they are accumulated only over a well defined time period ∆τ whichwill be asumed to be much shorter than the observation time ∆t. If the radi-ation flux consists of rℓ incoherent wave trains of type ℓ per unit time, eachof the wave packets can induce the transition independently of the other ones(they are incoherent!) and the probabilities simply add up (the reasoning isexactly the same as the one we have used to derive the formula (3.26) for

63

I(ω)).The constant transition probability per unit time is in this situationequal

wind(i→ f) =∑

ℓ

rℓ |a(1)fi [Aℓ]|2 . (3.30)

These transition probabilities (per unit time) can be then expressed throughthe incoming flux spectral density (3.26):

wind(i→ f) =4π2

~2c2c I(ωfi) |ǫ(k)·p(+)

fi |2 if Ef > Ei , (3.31)

wind(i→ f) =4π2

~2c2c I(ωif) |ǫ∗(k)·p(−)

fi |2 if Ef < Ei , (3.32)

In our simple model of the electrons in the atom (spin neglected, inter-actions between electrons taken into account using the mean field approxi-mation) it is straightforward to compare the rates of the transitions f → iand i → f . Assuming that Ef > Ei, this reduces to comparing the factor

ǫ∗(k)·p(−)

if with ǫ(k)·p(+)fi : using the definition (3.29) we write

ǫ∗(k)·p(−)

if =e~

Mωif

∫

d3ru∗i (r) e−ik·r

ǫ∗ ·∇uf(r)

= − e~

Mωif

∫

d3ruf(r) ǫ∗ ·∇

(

e−ik·r u∗i (r))

=e~

Mωfi

∫

d3ruf(r) e−ik·r

ǫ∗ ·∇u∗i (r) =

(

ǫ(k)·p(+)fi

)∗

. (3.33)

In the first step we have integrated by parts and used the fact that theboundary term vanishes because uf and ui are the wave functions of boundstates. In the second step we have used the relation (3.15). Thus,

wind(i→ f) = wind(f → i) . (3.34)

In most applications the wavelength λ related to the frequency ωfi corre-sponding to the considered atomic transition is much larger than the typicalatomic size set by the Bohr radius aB = ~

2/Me2 (beyond which the wavefunctions of bound states practically vanish).7 In such cases, because the fac-tors exp(±ik·r) in (3.29) are then close to unity (k·r < |k||r| ∼ aB/λ≪ 1),it makes sense to write

p(±)fi =

ie

Mωfi〈f | (1± ik·r− 1

2(k·r)2 + . . .)p |i〉 , (3.35)

7This is almost always true as far as transitions in the Hydrogen atom are concerned:|k| = ωif/c is always bounded by 13.6 eV/~c; since the n-th “Bohr” radius a (that isthe distance beyond which the wave functions of the |nlml〉 states practically vanish) isa = naB, |k|a <

∼ 3× 10−3 n and, therefore, |k|a ≪ 1 for n ≪ 1000.

64

and to base the classification of transions on the resulting multipole expansion

p(±)fi = (p

(±)fi )E1 + (p

(±)fi )M1 + (p

(±)fi )E2 + . . . , (3.36)

because the terms of (3.36) corresponding to higher powers of the r operatorin (3.35) are hierarchicaly suppressed.

The first term of (3.36), arising from the term of zeroth order in r in(3.35) is related to the atom’s electric dipole moment operator d = −er.Using the relation

p =iM

~[H0, r] , (3.37)

with H0 given by (3.20) one immediately gets that (ωfi = (Ef −Ei)/~)

(p(±)fi )E1 =

ie

Mωfi〈f |p|i〉 = −e 〈f |r|i〉 = dfi . (3.38)

The terms (p(±)fi )M1 and (p

(±)fi )E2 of (3.36) arise both from the term of

(3.35) linear in r after splitting its contraction with ǫ(n) in the followingway8 (recall, k = (|ωfi|/c)n = (±ωfi/c)n)

− e

2Mcnlǫj(n) 〈f | rlpj − rjpl |i〉 = ǫ(n)·(p(±)

fi )M1 , (3.39)

− e

2Mcnlǫj(n) 〈f | rlpj + rjpl |i〉 = ǫ(n)·(p(±)

fi )E2 . (3.40)

The first one can be represented in the form

ǫ(n)·(p(±)fi )M1 = (n×ǫ(n))·mfi , (3.41)

in which mif is the matrix element 〈f |m|i〉 of the atomic (orbital) magneticmoment operator9 (recall, we are using the Gauss’ units)

m = − e

2McL , (3.42)

8For simplicity of the notation we take the polarization vectors to be real here.9Matrix elements mj

fi of components of the vector operator m can be also represented

in terms of the atomic current density jlfi(r) = −e(~/2Mi)[u∗f(r)∂lui(r) − ∂lu

∗f (r)ui(r)]:

mnfi = − e

2c

∫

d3r (r×jfi(r))n = − e

2c

∫

d3r ǫnslrs[u∗

f (r)∂lui(r)− ∂lu∗

f (r)ui(r)]~

2Mi

= − e

2c

∫

d3r ǫnslrsu∗

f (r)∂lui(r)~

Mi≡ − e

2Mc〈f |Ln|i〉 .

Recall, that in the considered model of the atom spin is neglected and the magneticmoment can have only the orbital origin.

65

with L = r× p. Notice, that the form of (3.42) corresponds precisely to theclassical gyromagnetic relation. The second one can be rewritten using theidentity

rlpj + rjpl = iM

~[H0, r

lrj] + i~δlj . (3.43)

(which is easily proven by working out the commutator with the help of theformula (3.37)) in the form

ǫ(n)·(p(±)fi )E2 = − e

2Mcnlǫj(n) iMωfi〈f |rlrj|i〉 =

i

6

ωfi

cnlǫj(n)Qlj

fi , (3.44)

where Qlj = −e (3 rlrj − r2δlj) is the electric quadrupole moment operator.10

If for a given pair of states |i〉 and |f〉 the matrix element dfi is notzero, the first term of the expansion (3.36) dominates and the correspondingtransition is called electric dipole transition (or allowed). Probabilities perunit time of such transitions caused by the radiation flux characterized bythe spectral density I(ω) and coming from the direction n are given by

w1E

ind(i→ f) =4π2

~αEM I(|ωfi|) |ǫ(n)·〈f |r|i〉|2 . (3.45)

It is easy to see from the formula (3.45) and the properties of the spher-ical harmonics, that such transitions can occur (the matrix element dfi isnonzero) only if the angular momentum of the electron in the atom changesby one unit: |∆L| = 1. The change ∆ml of the magnetic quantum numbercan take the values 0 and ±1 depending on the direction of the polarizationvector ǫ(n).

Transitions i→ f , for which the relevant matrix element of the operatorr vanishes are called forbidden. They can have nonzero probabilities (perunit time) if the relevant matrix elements of the magnetic moment (theseare called magnetic dipole, or M1, transitions) or of the electric quadrupolemoment operators (E2 transitions) are nonzero. Their transitions rates arethen given by

w1M

ind (i→ f) =4π2

~2cI(|ωfi|) |n×ǫ(n))·mfi|2 , (3.46)

w2E

ind(i→ f) =4π2

~2c3ω2fiI(|ωfi|)

∣

∣

∣

∣

1

6nlǫj(n)Qlj

fi

∣

∣

∣

∣

2

. (3.47)

and are suppressed by a factor of order ∼ (aB/λ)2 ≪ 1 compared to the elec-

tric dipole ones. Higher order multipole transitions are yet more suppressed.

10The δlj term of (3.43) does not contribute due to the the relation n · ǫ(n) = 0. The

same circumstance allows to complete the operator 3 rlrj to Qljfi.

66

If both, initial and final states are spherically symmetric (S-states) thenthe whole matrix element in the exact expressions (3.31) and (3.32) van-ish.11 Such transitions are called strictly forbidden. Their probabilities canbe, however, nonvanishing if the term ∝ A2, neglected in Vint given by (3.21),is taken into account. In the framework of quantum electrodynamics suchtransitions correspond to the absorption/emission of two or more photons.Their rates are highly suppressed compared to one-photon transitions (typi-cally by a factor of order 108). This statement is no longer true if spin andrelativistic effects are taken into account but the single photon transition in-duced by these effects between S states are still more suppressed with respectto the two-photon transitions (by factors of order 106).

If the atom is placed in the black body radiation in a cavity where it isirradiated from all possible directions and the polarizations of wave trains arerandom, the quantity of interest are the transition probabilities wind(i→ f)averaged first (for fixed direction n) over the two possible polarization vectorsǫ(n, λ) and next over all possible directions n of the wave vector k. This is

easily done in the dipole approximation in which the factors p(±)fi do not

depend of n and the dependence on the direction enters only through thepolarization vectors. Averaging over polarization can be performed usingthe relation

∑

λ=±1

ǫi(k, λ)·ǫj∗(k, λ) = δij − kikj

k2= δij − ninj , (3.48)

(it follows from the fact that the triplet of the unit vectors: ǫ(k,+1), ǫ(k,−1)and n, spans the whole three-dimensional space). This gives

1

2

∑

λ=±1

|ǫ(n)·〈f |r|i〉|2 = 1

2

(

|〈f |r|i〉|2 − |n·〈f |r|i〉|2)

.

Averaging over the propagation directions n of the radiation is performedwith the help of the formula12

1

4π

∫

dΩn ninj =

1

3δij , (3.49)

11This can be seen as follows: if the z axis is taken along the vector k, the relevantmatrix element in (3.29) is proportional to

∫

d3ru∗

f (r) e±ikz

ǫ·∇ui(r) .

If the wave functions u∗f (r) and ui(r) are spherically symmetric, the the integrand vanishes

being an odd function of x and y (since in view of (3.15) ǫ·∇ = ǫx∇x + ǫ

y∇y).12The formula (3.49) can be directly checked by taking ni = (sin θ cosφ, sin θ sinφ, cos θ)

and dΩn = dφdθ sin θ, but the quicker argument is that its right hand side must beproportional to δij (there is no other symmetric two-index tensor available); the coefficientcan be fixed by contracting both sides with δij (using δijninj = n2 = 1).

67

which leads to the result

1

4π

∫

dΩn

1

2

∑

λ=±1

|ǫ(n)·〈f |r|i〉|2 = 1

3〈f |r|i〉·〈f |r|i〉∗ ≡ 1

3|〈f |r|i〉|2 . (3.50)

Thus, in the electric dipole approximation the probability per unit time thatan atom placed in the cavity filled with black body radiation characterizedby the spectral density ρ(ω) = I(ω)/c (which is in this context a more appro-priate quantity than the flux spectral density I(ω)) will make the transitionfrom the stationary state |i〉 to another stationary state |f〉

w1E

ind(i→ f) = w1E

ind(f → i) =4π2c

3~αEM ρ(|ωfi|) |〈f |r|i〉|2 . (3.51)

Expressed in this way the formula for w1E (i → f) is valid in both, Gaussand SI, systems of units.

3.3 Spontaneous emission

In classical physics a charged harmonic oscillator interacting with the exter-nal field of an electromagnetic plane wave absorbs or emits energy dependingon the relation between the phase of its motion and the phase of the radi-ation field. This effect is analogous to the induced transitions discussed inthe preceding subsection. A charged oscillator emits energy also in the ab-sence of an external electromagnetic field, because according to the classicalMaxwell theory any charged particle must radiate when it is accelerated. Inthis section we first recall the formula for the energy emitted by a classicalsource and then translate it into quantum language using an old Bohr-stylearguments. Application of the resulting expression for the probability perunit time of spontaneous atomic transitions, combined with the ones of in-duced transitions (derived in section 3.2), to the matter-radiation equilibriumwill provide, via the Planck law, the check of the correctness of the heuristicarguments.

We begin by considering a classical current J(t, r). The time evolution ofthe charge density ρ is not independent - it is determined by the continuityequation (3.3) once its space distribution at some initial time t0 is specified.Given the current J(t, r), the magnetic B(t, r) and the electric E(t, r) fieldscan be determined from the left pair of the Maxwell equations (3.1) and (3.2).Taking the rotation of both sides of

∇×B− 1

c

∂E

∂t=

4π

cJ , (3.52)

68

and using the fact that in the Cartesian coordinates ∇×∇×B = ∇(∇ ·B)−∇

2B, as well as the Maxwell equation

∇×E = −1

c

∂B

∂t, (3.53)

one arrives at the wave equation satisfied by each component of B:

1

c2∂2B

∂t2−∇

2B =4π

c∇×J . (3.54)

Let us assume that the time dependence of the current J(t, r) is harmonic:

J(t, r) = J(r) e−iωt + J∗(r) e+iωt ≡ J(r) e−iωt + c.c. (3.55)

We will also assume that the current J(r) is localized in a small region ofspace around r = 0 (in application to the spontaneous emission of atomswe will assume that J(r) vanishes for |r| big compared to the Bohr radiusaB ∼ 0.5 ·10−10m). The electric and magnetic fields must have the same timedependence13 as J(t, r):

B(t, r) = B(r) e−iωt + c.c. , (3.56)

E(t, r) = E(r) e−iωt + c.c. . (3.57)

The wave equation (3.54) for B(r) then becomes

(

∇2 + k2

)

B(r) = −4π

c∇×J(r) , (3.58)

where k = ω/c. This is solved by using the Green’s function G(r − r′)satisfying

(

∇2 + k2

)

G(r− r′) = −δ(3)(r− r′) . (3.59)

The solution of the form of outgoing wave (obtained by the proper choice ofthe integration contour) has the well known form

G(r− r′) =1

4π

eik|r−r′|

|r− r′| . (3.60)

For future use we also record the following expansion

eik|r−r′|

|r− r′| ≈1

r

(

1 +r′ ·n+ i

2k(r′2 − |r′ ·n|2)r

+ . . .

)

eik(r−r′·n) , (3.61)

13The r dependent time delay - since the equation (3.54) has a relativistic form, changesof the current J localized e mear r = 0 affect the fields B(r) and E(r) at a distant pointr only after the time t ≈ |r|/c - is accounted for in the phases of the time independentcomplex vectors B(r) and E(r).

69

where14 r = |r|, n = r/|r|, and we have assumed that r ≫ |r′|. In terms ofthe Green’s function (3.60) the solution of the equation (3.58) takes the form

B(r) =1

c

∫

d3r′eik|r−r′|

|r− r′| ∇′×J(r′) , (3.62)

where the prime over ∇ means differentiation with respect to r′.

To determine the energy emitted per unit time by the oscillating current(3.55) we have to find the Poynting vector

P =c

4πE×B , (3.63)

at distances large compared to the spatial size of the current distribution. Atsuch distances the electric field E(t, r) is given by the equation (3.52) withJ = 0, which, taking into account the form (3.57) of E(t, r), implies that

E(r) = ic

ω∇×B(r) . (3.64)

Averaging the Poynting vector

P =c

4π

[

E(r) e−iωt + c.c.]

×[

B(r) e−iωt + c.c.]

, (3.65)

over the period T = 2π/ω we get

〈P〉T =c

2πRe [E(r)×B∗(r)] . (3.66)

We must find the leading (order 1/r) terms of both fields, E(r) and B(r),for r large compared to the spatial size of the current J(r) distribution. Thiswill give the 1/r2 term of the Poynting vector; the total energy emitted perunit time in the whole solid angle, given by the integral

∫

ds · 〈P〉T over alarge sphere of radius R → ∞, will be then independent of R. Let us findB(r) first. The expression (3.62) integrated by parts gives (the boundaryterm vanishes due to the spatial localization of J(r)):

B(r) =1

c

∫

d3r′ J(r′)×∇′

(

eik|r−r′|

|r− r′|

)

=1

c

∫

d3r′ J(r′)×∇′

(

eikr

re−ikn·r′ + . . .

)

(3.67)

=ik

c

eikr

rn×

∫

d3r′ J(r′) e−ikn·r′ +O(1/r2)

=ik

c

eikr

rn×J0(n) +O(1/r2) ,

14Note that n defined here as r/|r| acquires the meaning of the direction k/|k| of thewave vector k of the wave registered by the detector located at r.

70

where

J0(n) ≡∫

d3r J(r) e−ikn·r . (3.68)

In the similar manner we compute E(r) from (3.64):

E(r) =i

kc

∫

d3r′∇×[

J(r′)×∇′

(

eik|r−r′|

|r− r′|

)]

. (3.69)

Writing this in components

Ei(r) =i

kc

∫

d3r′ ǫijkǫklm Jl(r′) ∂j∂′m

(

eik|r−r′|

|r− r′|

)

= − i

kc

∫

d3r′ ǫijkǫklm Jl(r′) ∂′j∂′m

(

eik|r−r′|

|r− r′|

)

(3.70)

= − i

kc

eikr

rǫijkǫklm

∫

d3r′ Jl(r′) ∂′j∂′m

(

e−ikn·r′ + . . .)

we find

E(r) = −ikc

eikr

rn×[n×J0(n)] +O(1/r2) . (3.71)

The time averaged Poynting vector (3.66) can be now computed with thehelp of a straightforward vector algebra

〈P〉T =k2

2πc

1

r2n(

|J0(n)|2 − |n·J0(n)|2)

+O(1/r3) . (3.72)

The formula (3.72) can be given the following interpretation: the energyflow has the radial direction, 〈P〉T ∝ n, and at a given observation point,to which the vector n points, the intensity of the radiation is proportionalto the length of the projection of J0(n) onto the plane perpendicular to thedirection of sight.

If the wavelength λ = 2π/k of the radiation produced is large compared tothe size of the spatial region to which J(r) is confined, one can approximatethe factor exp(−ikn · r′) appearing under the integrals over d3r′ in (3.67)and (3.70) by unity. This corresponds to the dipole approximation. In thisapproximation the quantity J0(n), which will be denoted JE1 , is independentof the direction n.

The formulae derived here can be carried over to quantum mechanics todescribe the radiation of frequency ω = ωif = (Ei − Ef )/~ emitted in thespontaneous transition of the atom from a state i to a lower state f (i.e.when Ei > Ef), provided one knows what to substitute for the current J(r).

71

It is clear that this current has to depend somehow on the initial and finalatomic states. We will assume that in quantum mechanics it is given by

J(r) = − e~

iMu∗f(r)∇ui(r) , (3.73)

and the frequency in (3.55) is just ωkn. The motivation for this choice(guess) is provided by the Bohr’s correspondence principle: classically J(r) =−ev(t, r), where v = p/M is the electron velocity. Therefore, the expression(3.73) can be taken for the electric current associated with the transitionfrom a state |k〉 to a state |n〉. The vector J0(n) should be then replaced byJfi(n)

Jfi(n) = −∫

d3re~

iMu∗n(r)∇uk(r) e

−ikn·r . (3.74)

In the dipole approximation (exp(−ikn·r) = 1 under the integral) this sim-plifies to

Jfi(n) ≈ − e

M〈f |p|i〉 = −ie ωfi 〈f |r|i〉 ≡ −iωifdfi ≡ JE1

fi , (3.75)

where we have used the relation (3.37).

The power of the radiation, that is, the total energy emitted per unittime (given by the integral

∫

ds · 〈P〉T over a large sphere of radius R withR → ∞) can be easily computed in the dipole approximation because JE1

fi isindependent of the direction n. Integrating the energy flux (3.72) using theformula (3.49) gives

〈power〉T =4k2

3c

∣

∣JE1

fi

∣

∣

2.

Thus, in this approximation, the power emitted by the atom which makesthe transition i→ f is

〈power〉T =4e2ω4

if

3c3|〈f |r|i〉|2 = 4~ω4

if

3c2αEM |〈f |r|i〉|2 . (3.76)

In the quantum theory the formula (3.76) for 〈power〉T is interpreted as

〈power〉T = number of spontaneous transitions in unit time× (Ei −Ef ) ,

with the assumption that in each transition one quantum of radiation withfrequency ω = (Ei−Ef )/~ and carrying away the energy Ei−Ef is emitted.On the other hand, the number of spontaneous transitions per unit time is(because we are considering a single atom) just the transition probability

72

per unit time w(i → f). Thus, in the dipole approximation we obtain theformula:

w1E

spont(i→ f) =4e2ω3

if

3~c3|〈f |r|i〉|2 =

4ω3if

3c2αEM |〈f |r|i〉|2 . (3.77)

(We have used Ei − Ef = ~ωif).

The sum of the probabilities of all spontaneous transitions from a givenatomic state |i〉 determines its width and lifetime. The total width of a state|i〉 is given by

Γi =∑

Ef<Ei

Γfi ≡∑

Ef<Ei

~wspont(i→ f) , (3.78)

where Γfi are the partial widths corresponding to individual transitions. Thetotal widths of the state |i〉 is related to its lifetime by

τi =~

Γi. (3.79)

Therefore, to find the lifetime of an atomic (quasi-)stationary state one hasto calculate probabilities of spontaneous transitions to all lower states (or atleast to those which correspond to largest probabilities). The finite lifetimeof a state inevitably leads to its nonzero width. Indeed, the uncertaintyprinciple for time and energy, ∆t∆E ∼ ~ tells us that in order to measurethe energy of a state with precision ∆E, the measurement has to take timenot shorter than ~/∆E. If the electron spends in the state only the time(on average) τ , its energy cannot be determined with precision better than∆E ∼ ~/τ = Γ. Typical lifetimes of the atomic states from which transitionsto lower states are allowed in the dipole approximation are of order 10−(8÷9)

sec., and grow with the principal quantum number n approximately as n9/2

(Bethe & Salpeter, 1957).

As an example let us determine the lifetime of the 2P states of the Hy-drogen atom. To this end we need to compute the matrix element

〈1S|r|2Pml〉 =∫ ∞

0

dr r2∫

dΩR∗10(r)Y

∗00(θ, φ) rR21(r)Y1ml

(θ, φ) .

Using the radial wave functions

R10(r) =2

√

a3Bexp(−r/aB) ,

R21(r) =1

2√

6a3B

r

aBexp(−r/2aB) ,

73

in which aB = ~2/Me2 we find

∫ ∞

0

dr r2R∗10(r) r R21(r) = 4

√6aB

(

2

3

)5

.

To work out the angular part we write the three Cartesian components ofn ≡ r/|r| as 1

2sin θ(e−iφ + e+iφ), − i

2sin θ(e−iφ − e+iφ) and cos θ, respectively

and use the spherical harmonics

Y1,1 = −√

3

8πeiφ sin θ , Y1,−1 =

√

3

8πe−iφ sin θ , Y1,0 =

√

3

4πcos θ .

This gives

∫

dΩY ∗00 nx Y1,ml

= −ml√6,

∫

dΩY ∗00 ny Y1,ml

= −i |ml|√6,

∫

dΩY ∗00 nz Y1,ml

=1√3δml0 .

The factor |〈1S|r|2Pml〉|2 ≡ 〈1S|r|2Pml〉∗ · 〈1S|r|2Pml〉 is independent ofthe magnetic number ml and equals

|〈1S|r|2Pml〉|2 = (4√6aB)

2

(

2

3

)101

3= 25a2B

(

2

3

)10

.

From the general formula (3.77) with ~ωif = 12Mc2α2

EM(1 − 14) = 3

8Mc2α2

EM

we find

wspont(2P → 1S) =4

3

(

3

8

Mc2α2EM

~

)3

αEM

(aBc

)2 215

310=

(

2

3

)8Mc2

~α5EM ,

that is, wspont(2P → 1S) = 0.63× 109 sec−1. The lifetime of each of the 2Pstates of the Hydrogen atom is therefore τ2P = 1.59× 10−9 sec.

3.4 Spectrum of the spontaneous radiation

In the semiclassical approximation to discuss the time profile of the electro-magnetic wave radiated spontaneously by an atom one has to assume thatthe atom got excited at t = 0 by some interaction with the environment(e.g. thermally). One can then assume that it is in a nonstationary state

74

whose wave function is the superposition15 (for simplicity we consider onlysuperpositions of two atomic states):

ψ(t) = aie−iEit/~ui(r) + afe

−iEf t/~uf(r) , (3.80)

with |ai|2 + |af |2 = 1. In a gas of atoms prepared in the same way the wavefunction (3.80) can be given a simple statistical interpretation. Namely, thefactors |ai|2 and |af |2 represent the fractions of the total number of atomsaccupying the states |i〉 and |f〉, respectively. The mean atom energy is then

〈E〉 = |ai|2Ei + |af |2Ef . (3.81)

Since the atom is supposed to radiate, its mean energy should decrease withtime. Hence, the factors |ai|2 and |af |2 cannot be strictly constant; we mustadmit they are slowly varying functions on the time scale set by 2π~/(Ei −Ef ).

The probability current corresponding to the wave function (3.80) uponmultiplication by the electric charge −e of the electron can be interpretedas the density of the electromagnetic current generated by the atom in thenonstationary state16 (3.80):

J(t, r) = − e~

2Mi

(

u∗f∇ui −∇u∗f ui)

e−iωif t a∗fai + c.c. (3.82)

In the expressions like (3.67) and (3.71), in which one can integrate by partsthe expression (3.68) for J0(n) (because differentiation of the exponent un-der the integral in (3.68) gives zero owing to the orthogonality of tha wavefunctions ui and uf) one effectively gets for the current the expression (3.73)multiplied by a∗nak. Thus, the expression (3.76) giving in the dipole approx-imation the emitted power (averaged over the period) is replaced by

〈power〉T =4k2

3c

∣

∣JE1

fi

∣

∣

2 ∣∣a∗fai

∣

∣

2, (3.83)

with JE1

fi given by (3.75).

Obviously, this emitted power should be equal to the atom’s averageenergy loss per unit time

d

dt

(

|ai|2Ei + |af |2Ef

)

= −〈power〉T . (3.84)

15If the considered atom, initially, say in the ground state, collides with another atom,its internal state after the collision is a superposition of all possible stationary states i withthe coefficients ai, which are, in principle, calculable, given the dynamics of the atomiccollisions (the interaction Hamiltonian).

16The current density corresponding to the wave function (3.80) has also two terms notproportional to e±iωif t; these do not produce the radiaton field (i.e. the field falling off atlarge distances only as 1/r).

75

Combined with the normalization condition |af |2 = 1 − |ai|2 and the con-stancy of Ei and Ef this leads to the differential equation for the time de-pendence of |ai|2:

d

dt|ai|2 = − 1

~ωif

4k2

3c

∣

∣JE1

fi

∣

∣

2 |ai|2(

1− |ai|2)

, (3.85)

or, using the definition (3.78),

d

dt|ai|2 = −Γfi

~|ai|2

(

1− |ai|2)

. (3.86)

The solution is

|ai(t)|2 =[

1 +1− |ai(0)|2|ai(0)|2

eΓnk~

t

]−1

, |af(t)|2 = 1− |ai(t)|2 . (3.87)

If only a small fraction of atoms is excited, i.e. if |ai(0)|2 ≪ 1, so that onecan approximate

|ai(t)| ≈ |ai(0)| e−Γfi2~

t , |af(t)| ≈ 1 , (3.88)

the factor Γfi is measured as the width of the frequency spectrum of theemitted radiation. To see this we need to compute anew the Poynting vec-tor of the radiation generated by the current (3.82). Since ai(t)a

∗f (t) is a

slowly varying factor compared to the rapid oscillations of the current withfrequency ωif , the time dependent magnetic and electric fields are still to agood accuracy given by (3.56) and (3.57) with B(r) and E(r) given by (3.67)

and (3.71) in which J0 is replaced by J0ai(t)a∗f (t) ≈ |ai(0)|J0 e

−Γfi2~

te−iφ′

where φ′ is the phase of ai(t)a∗f(t) which is unimportant. The expression

(3.63) for the Poynting vector (now not averaged over the period) gives then

n·P ∝ −e+2ikr[

JE1

fi ·JE1

fi − (n·JE1

fi )2]

e−2iφ′

e−2iωknte−Γnk~

t

−e−2ikr[

JE1

fi ·JE1

fi − (n·JE1

fi )2]∗e+2iφ′

e+2iωknte−Γnk~

t

+2[

JE1

fi ·JE1∗fi − (n·JE1

fi )(n·JE1∗fi )

]

e−Γnk~

t (3.89)

If the atomic wave functions ui and uf in (3.80) are taken to be real, allcomponents of the current vector J0 have the same phase and the moduli ofthe square brackets in the above formula are all equal. As we are interestedonly in the time dependence of the energy flow, at a fixed distance r at whichthe measuring devce is place we can write

n·P ∝[

2− e−2iωif t−2iφ − e2iωif t+2iφ]

e−Γnkt/~

∝[

e−Γif t/2~ sin(ωif t + φ)]2, (3.90)

76

with φ being the sum of φ′, of the phase resulting from exp(−ikr) and of thephase of J0. Since the atom is assumed to have gotten excited at t = 0, thisformula gives the energy registerd by the measuring device only for t > 0;for t ≤ 0, the Poynting vector vanishes. The spectral shape Iemitted(ω) of theatom’s radiation can be determined from the formula (which equates two theforms of writing the total energy emitted)

∫ ∞

0

dω Iemitted(ω) =

∫ ∞

−∞

dtn·P(t) ≡∫ ∞

0

dtn·P(t) . (3.91)

The right hand side can be written applying the Parseval’s identity to thereal function f(t) (so that its Fourier transform f(ω) satisfies the relationf ∗(ω) = f(−ω))

∫ ∞

−∞

dt |f(t)|2 =∫ ∞

−∞

dω

2π|f(ω)|2 =

∫ ∞

0

dω

π|f(ω)|2.

With

f(t) =

0 if t < 0e−Γif t/2~ sin(ωif t + φ) if t > 0

(3.92)

one finds

f(ω) =1

2

eiφ

ωif + ω + iΓif/2~+

1

2

e−iφ

ωif − ω − iΓif/2~. (3.93)

Thus the spectral shape Iemitted(ω) ∝ |f(ω)|2 of the light emitted by the atomin the spontaneous transition from the state |i〉 to the state |f〉 has a sharppeak at ω = ωif which is approximated by

Iemitted(ω) ∝1

(ω − ωif)2 + (Γif/2~)2. (3.94)

Just these peaks are observed as the spectral lines characteristic for atomsof a given element (they form the so-called linear spectra).

Experimentally the formula (3.94) is true only if |f〉 is the atom’s groundstate (which is absolutely stable). If |f〉 is an excited state (of energy lowerthan the energy if |i〉), there is additional broadening of the spectral line.This was explained by V. Weisskopf who using the full fledged quantumelectrodynamics, showed that in this case Γif in (3.94) has to be replaced byΓif + Γf , where Γf is the total width of the final state |f〉.

The relative intensities of spectral lines in the light emitted by a gas ofexcited atoms of a given element, that is the relative heights of the peaksin the spectral profile of the emitted light Iemitted(ω), are determined by the

77

powers emitted at frequencies ωif . The power emitted at the given frequencyωif in transitions between two concrete states |i〉 and |f〉 is given by (in thedipole approximation) by the formula (3.76). Since there can be differentpairs of states giving rise to the same frequency ωif , the right hand side of(3.76) has to besummed over all such final states |f〉 and averaged over allinitial states |i〉. The intensity of the rediation is therefore proportional tothe factor

Ifi = total power emitted at frequency ωif =4

3

~ω4if

c2αEM

Sfi

gi, (3.95)

where the line strenght Sfi is defined as

Sfi =

gf∑

b

gi∑

a

|〈fb|r|ia〉|2 , (3.96)

and gi and gf are multiplicities of the degenerate initial Ei and final Ef

energy levels, respectively.

3.5 Einstein’s coefficients

The correctness of the formula (3.77) for the probability of spontaneous tran-sitions per unit time (which we have derived by using some hand-wavingarguments) can be verified by the following reasoning. Consider the radia-tion at the temperature T in a cavity. Let us assume that the radiation isin thermal equilibrium with the cavity’s walls. It is constantly emitted andabsorbed by atoms in the walls of the cavity. Let us assume that each partic-ular frequency of the radiation corresponds to the energy difference (dividedby ~) of some pair of atomic energy levels. Thus absorption of the radiationquanta of a frequency ω is due to induced transitions from atomic states ofenergy En to states of higher energy Ek such that Ek − En = ~ω, while theemission of the radiation quanta of this frequency is due to two mechanisms:induced and spontaneous transions from states of energy Ek to states of en-ergy En. In equilibrium the number of radiation quanta of (any) frequencyω in the cavity does not change in time. This means that, per unit time, thenumber of induced transitions from the atomic states corresponding to theenergy level En to the state at the level Ek should be equal to the numberof induced and spontaneous transitions from the states having the energy Ek

to the states of energy En. Therefore, (working in the dipole approximation)the following equality should hold

Nkn4π2c

3~αEM ρ(ωkn) |〈k|r|n〉|2

= Nnk

(

4π2c

3~αEM ρ(ωkn) |〈n|r|k〉|2 +

4ω3kn

3c2α |〈n|r|k〉|2

)

. (3.97)

78

where Nkn and Nnk are the appropriate statistical factors which we nowcompute.17

The walls of the cavity treated as a single quantum system contain anumber N of electrons which can occupy many of the available one-particlestates. There are many similar one-particle states centered on different atomsso there is a huge degeneracy. The number of electron transitions from theenergy level En to the energy level Ek is proportional18 to the numbers gnand gk of states with energies En and Ek, respectively, and to the averageoccupation number nn of an individual one-particle state of energy En whichis given by the Fermi-Dirac distribution

nn =(

eEn−µkBT + 1

)−1

, (3.98)

where kB is the Boltzmann constant and µ is the chemical potential of elec-trons (determined by the condition that the total number of electrons is N).In addition, since electrons are fermions, the transition cannot occur to acompletely occupied state. Hence, the number of electron transitions fromthe state |n〉 to the state |k〉 must be also proportional to

1− nk = 1−(

eEk−µ

kBT + 1

)−1

=e

Ek−µ

kBT

eEk−µ

kBT + 1. (3.99)

Similar reasoning applies to transitions in the opposite direction. Thus

Nkn = gngknn(1− nk) = gngke

Ek−µ

kBT

[eEk−µ

kBT + 1][eEn−µkBT + 1]

Nnk = gkgnnn(1− nk) = gkgne

En−µkBT

[eEn−µkBT + 1][e

Ek−µ

kBT + 1]. (3.100)

Inserting the factors (3.100) into the detailed balance equation (3.97) weget

e−En/kBT×4π2c

3~αEM ρ(ωkn) |〈k|r|n〉|2

= e−Ek/kBT×(

4π2c

3~αEM ρ(ωkn) |〈n|r|k〉|2 +

4ω3kn

3c2αEM |〈n|r|k〉|2

)

. (3.101)

17We follow here the exposition of R.N. Zitter & R.C. Hilborn, Am. J. Phys. 55 (1987)p. 522; I thank prof. M. Napiorkowski for bringing this reference to my attention.

18The tacit assumption we are making in the balance equation (3.97) is that the transi-tion probabilities per unit time depend only on the energy levels and not on the choice ofparticular states from these levels. This is true because we use the transition probabilitiesaveraged over the polarization of the emitted/absorbed radiation and over the directionsof its wave vector k.

79

(the denominators of the factors (3.100) and the degeneracy factors gkgnhave canceled on both sides). The balance equation in this form would alsofollow if we simply assumed, as is usually done in textbooks (and as Einsteinoriginally did), that the number of transitions from the n-th energy level tothe k-th level is proportional to the number gk of states forming the k-thlevel and to the number of atoms in the state corresponding to the n-th levelwhich in turn (in classical statistical physics) is gven by the the Boltzmannformula

Nn = Z−1statgn exp

(

− En

kBT

)

, (3.102)

in which Zstat is the normalization factor (the statistical sum). Applyingclassical statistical physics is of course in line with the semiclassical approachto the radiation theory of atoms but it is reassuring to see that the balanceequation does not depend at this point on the classical approximation.19

Taking into account that |〈n|r|k〉|2 = |〈k|r|n〉|2 we obtain from the rela-tion (3.97) the formula

ρ(ω) =~ω3

π2c3 (e~ω/kBT − 1)(3.103)

This agrees with the celebrated Planck formula for the energy density ρ(ω)of the black body radiation. This agreement constitutes a nontrivial checkof the formula (3.77) for the probability of spontaneous emission derivedin the preceding subsection. We also see that spontaneous emissions andthermal equilibrium are intimately related to each other: the possibility ofestablishing the latter requires that there must be spontaneous emissions.

As a historical remark, let us remind that in the years 1916-1917 A.Einstein considered spontaneous and induced transitions in thermal equilib-rium and introduced the coefficients Bnk characterizing the probability of

19If the electrons were bosons, the number of electron transitions from the n-th level tothe k-th level would be proportional to the numbers gn and gk of states with energies En

and Ek, respectively, and to the average occupation number nn of a single state of energyEn which in this case would be given by the Bose-Einstein distribution

nn =(

eEn−µkBT − 1

)−1

.

In addition, one would then have to take into account that the transition probability tothe k-th state which is occupied by nk bosons is bigger by the factor nk + 1 than thetransition probability to an empty state. Hence, one would have

Nkn = gkgnnn(1 + nk)

and one would end up again with the same form (3.101) of the balance equation.

80

the induced transition from an individual state of energy Ek to any of thestates20 from the energy level En and Ank characterizing in the same waythe probability of the spontaneous transition (which can occur if Ek > En).Assuming that probabilities of induced transitions are proportional to the ra-diation spectral energy density ρ(|ωkn|) of the radiaton and that transitionsbetween two energy levels occur only via the emission or absorption of a lightquantum of frequency ω = |Ek −En|/~ Einstein wrote the thermodynamicalequilibrium condition similar to (3.97) (we assume Ek > En):

BknNnρ(ω) = BnkNkρ(ω) + AnkNk , (3.104)

with Nn and Nk given by (3.102). After rewriting it in the form

Ankgk = ρ(ω) [Bkngn exp((Ek − En)/kBT )− Bnkgk] , (3.105)

he noticed that if this formula is to be consistent at high temperatures Twith the Rayleigh-Jeans law (derived by these authors in classical electrody-namics), which states that the radiation energy density ρ(ω) is proportionalto T , the equality Bkngn = Bnkgk must hold because Ank does not depend inT . In this way, he arrived at the relation21

ρ(ω) =Ank/Bnk

e(En−Ek)/kBT − 1=

Ank/Bnk

e~ω/kBT − 1, (3.106)

and by comparing it with the (experimentally confirmed) Planck formulaconcuded that

Ank =~ω3

π2c3Bnk . (3.107)

(Note that dividing the probability w1E

spont(k → n) of the spontaneous transi-tions (3.77) by the corresponding probability of the induced transition (3.51)divided by ρ(ω) we indeed get ~ω3/π2c3). The formula (3.107) can be usedto derive the expression for the probability (per unit time) of spontaneousemission from the coefficient of the induced transitions (whose derivation inthe semiclassical approach relies on more solid ground) and the assumptionthat the thermal equilibrium of radiation can be establihed. Historically theEinstein’s reasoning also showed that the Planck derivation of the formulafor the energy density of the black body radiation did not in fact relied onthe equal spacing of energy levels of the harmonic oscillator (which Planckhad used in his derivation). It provided also the first link between the Planckblack body radiation formula with the Bohr’s theory of atomic spectra andwas the first application of a probabilistic reasoning to quantum physics.

20Notice that the multiplicity factors gn are in the Einstein’s definition included in thecoefficients Bnk and Ank.

21Notice that without the second term on the right hand side of (3.105), i.e. withoutinduced transitions to lower energy states, one would obtain ρ(ω) as predicted by Wien.

81

3.6 Angular momentum

To complete the semiclassical considerations of spontaneous emissions we willnow calculate the angular momentum carried away by the emitted radiation.This will provide further check of the expression (3.73) for the quantumversion of the electric current.

Consider first the momentum density of the radiation. Its energy flux isgiven by the Poynting vector P (3.63). The radiation momentum density istherefore P/c2 because (even classically) the energy density of the radiationand its momentum density differ by the factor of c. Consequently, the angularmomentum of the radiation absorbed in ∆t by the area |ds| is given by

c ·∆t · ds·(

r×P

c2

)

. (3.108)

Averaging over the period with the help of the formula (3.66) and writingr = rn we get the angular momentum dL absorbed by dσ per unit time

dL = r2dΩr

cn× c

2πRe (E×B∗)

=1

2πdΩ r3Re [E(n·B∗)−B∗(n·E)] . (3.109)

From the calculation of the energy emitted crried out in section 3.3, weknow that the leading, 1/r, terms in the expressions for B(r) and E(r) gen-erated by the oscillating current J are perpendicular to the vector n (see theexpressions (3.56) and (3.57)). This is consistent with the formula (3.109)because otherwise dL would grow as r at large distances r. Therefore, toobtain dL independent of r we have to use the 1/r2 term of B∗ together withthe leading, order 1/r, term in E in the first part (i.e. in E(n · B∗)) of theexpression (3.109) and the other way around in the second part of (3.109).

Finding the 1/r2 terms in B(r) and E(r) is straightforward but somewhattedious. For B(r) we start from from the first line of (3.67) and use the 1/r2

term in the expansion (3.61):

B(r)|1/r2 =eikr

cr2

∫

d3r′ J(r′)×∇′

[

n · r′ + i

2k(

r′2 − (n·r′)2)

]

e−ikn·r′ .

To find the angular momentum dL we need to compute only n · B and notB itself. This simplifies the task, because the action of ∇′ produces manyterms proportional to n which give zero in view of the fact that

n·∫

J× n = 0 .

82

The only nonvanishing term comes from the action of ∇′ on r′2. This gives

n·B|1/r2 =eikr

cr2n·

∫

d3r′ (J(r′)× ikr′) e−ikn·r′ +O(

1/r3)

. (3.110)

To find the 1/r2 term in n ·E we proceed in the similar way:

n·E|1/r2 = − i

ckniǫijkǫklm

∫

d3r′ Jl(r′) ∂′j∂′m

(

eik|r−r′|

|r− r′|

)

= − i

ck

eikr

r2ni ǫijkǫklm

∫

d3r′ Jl(r′)∂′j∂

′m

[

n·r′ + i

2k(

r′2 − (n·r′)2)

]

e−ikn·r′,

Again the nonvanishing result is obtained only by acting with ∂′j on r′2:

n · E|1/r2 =eikr

r2c

∫

d3r′ [2n · J(r′) + ik (r′ · J(r′)− (n · r′)(n · J(r′))] e−ikn·r′.

We note that in the dipole approximation, in which one sets k|r′| = 0, n ·Bvanishes (it has an explicit factor kr′ under the integral) and n · E reducesto

n · E|1/r2 =eikr

r2c

∫

d3r′ 2n·J(r′) = eikr

r2c2n·J0 . (3.111)

Since the 1/r term in B does not vanish, in the dipole approximation weobtain

dL = − 1

2πdΩ r3Re [B∗(n·E)] . (3.112)

Using the 1/r term in B from (3.67), the result (3.111) and the vector n inthe form

n = (sin θ cosφ, sin θ sinφ cos θ) ,

and upon explicit integration over dΩ we find the total angular momentumemitted in unit time by the oscillating current (3.55):

L =4k

3c2Re (iJ0×J∗

0) . (3.113)

In particular,

Lz =4k

3c2Re

[

i(

J0xJ∗0y − J0yJ

∗0x

)]

. (3.114)

We see that for Lz to be nonzero the components J0x and J0y must havedifferent phases. For example, if J0z = 0 and J0y = iJ0x then the component

83

Lz (the only one nonvanishing in this case) of the angular momentum emittedper unit time is

Lz =4k

3c2(

|J0x|2 + |J0y|2)

=4k

3c2|J0|2 . (3.115)

Since the energy emitted per unit time is given by (3.76) we get that

Lz emitted per unit time =power

ck=

power

ω

which means that if (per unit time) one quantum of energy ~ω is emitted,the angular momentum carried by it equals ~.

3.7 Selection rules

General rules allowing to determine which transitions (either stimulated orspontaneous) can occur and which cannot are called selection rules.

In the case of the single electron bound in the atom by the sphericallysymmetric potential V (r) the eigenstates of H0 = −(~2/2M)∇2 + V (r) canbe chosen as eigenstates of L2 and Lz. Electric dipole transitions betweentwo states |n, l,m〉 and |n′, l′, m′〉 occur if the matrix elements of the positionoperator r between these states is nonzero. Since

〈n′, l′, m′| z |n, l,m〉 ∝∫

dΩY ∗l′m′ cos θ Ylm . (3.116)

and cos θYlm = aYl+1,m+bYl−1,m (a and b are known coefficients), the matrixelement of the z operator is nonzero only if

l′ = l ± 1 and m′ = m. (3.117)

In turn, matrix elements of the x and y operators are given by the integralsof the type

〈n′, l′, m′| x± iy |n, l,m〉 ∝∫

dΩY ∗l′m′ sin θ e±iφ Ylm . (3.118)

Since sin θ e±iφ Ylm = c Yl+1,m±1 + dYl−1,m±1 (again with known coefficients cand d) it follows that the x+ iy matrix element is nonzero if

l′ = l ± 1 and m′ = m+ 1 , (3.119)

whereas nonvanishing of the x− iy operator matrix element requires

l′ = l ± 1 and m′ = m− 1 . (3.120)

84

Parity of atomic states is given by (−1)l, while the electron electric dipole(position) operator is odd with respect to the parity in each electric dipoletransition the parity of the atom changes.

We can also consider spontaneous emissions for J0y = iJ0x and J0z = 0 inwhich case, as we have found in section 3.6, the angular momentum carriedaway by the each emitted radiation quantum equals ~. From the formulae

J0x ∝ 〈n′, l′, m′| x |n, l,m〉 ∝∫

dΩY ∗l′m′ sin θ

(

e+iφ + e−iφ)

Ylm ,

J0y ∝ 〈n′, l′, m′| y |n, l,m〉 ∝ −i∫

dΩY ∗l′m′ sin θ

(

e+iφ − e−iφ)

Ylm ,

which follow from (3.75), we see that the equality J0y = iJ0x requires that theparts of the above integrals with e+iφ vanish. This is so if m′ = m−1, whichmeans, that as a result of the transition the value of Lz of the atom decreasesby ~. This is another confirmation of the guess (3.73) we have made in orderto apply the results of the classical radiation theory to quantum mechanicsof the spontaneous emission.

The selection rules outlined here apply directly also to many electronatoms provided the interaction betwen electrons can be neglected. The en-ergy levels are then those of the Hydrogen-like atom with individual electronsoccupying different one-particle energy levels. The atom’s wave function canbe then (leaving aside for the moment the problem of its antisymmetrization)chosen in the form

ψk1,...,kN (r1, . . . , rN) = uk1(r1) · . . . · ukN (rN) . (3.121)

Since the Hamiltonian describing the interactions of atomic electrons withthe radiation is of the form

Vint(r1, . . . , rN) = Vint(r1) + . . .+ Vint(rN) , (3.122)

its matrix elements batween different atomic states relevant for calculatingrates of spontaneous and induced transitions factorize into

〈k1, . . . , kN |Vint|n1, . . . , nN〉 = 〈k1|Vint(r1)|n1〉+ . . .+ 〈kN |Vint(rN)|nN〉 .(3.123)

As a result, to transitions of individual electrons the same selection rules asfor the Hydrogen atom apply.

Even if electrons in the atom cannot be treated as mutually noninteract-ing, the selection rules ∆L = ±1 and Pf · Pi = −1, where Pf and Pi arethe parities of the final and initial atomic states, respectively, still apply (theLaporte’s rule).

85

In more general cases, when the spin effects are also taken into account,the selection rules can be established with the help of the Wigner-Eckarttheorem (to be discussed in the next section). The clasification of transitionsinto electric dipole, electric quadrupole, magnetic dipole etc. is then veryconvenient as the operators corresponding to these transitions have well de-fined properties with respect to rotations (it is this fact that enables us touse the Wigner-Eckart theorem).

3.8 Quantum theory of radiation

The semiclassical theory outlined in the preceding sections of this chaptercan readily be improved in several directions. Firstly, spin of the chargedmatter particles (electrons) interacting with the radiation field can be easilytaken into account by replacing the nonrelativistic single particle quantummechanics based on the Schrodinger equation (3.8) with the one based onthe Pauli equation (recall, that q is the particle charge in units of e > 0)

i~∂

∂tΨ =

[

1

2M

(

−i~∇− qe

cA)2

− qe

2Mcg~

2σ ·B+ qeϕ+ V

]

Ψ ,

for the two-component wave function Ψ with the gyromagnetic factor g ap-propriate for the charged particle interacting with the radiation.22 This en-tails the replacement of the interaction operator (3.21) with

Vint = − qe

McA(t, r)·p− qe

2Mcg~

2σ ·(∇×A(t, r)) .

The only change in the formulae (3.38), (3.39 and (3.40)) for (p(±)fi )E1 ,

(p(±)fi )M1 and (p

(±)fi )E2 is the replacement of the magnetic moment opera-

tor m given by (3.42) by

m =qe

2Mc(L+ g s) ,

where s = (~/2)σ is the electron spin operator. Spin projection has then tobe accounted also in specifying initial and final atomic states |i〉 and |f〉 (onemay be inerested e.g. in computing transition probabilities averaged over thespin directions in the initial state and/or summed over spin directions of thefinal state).

22The electron factor g equals 2 (plus small corrections). This value is necessary toquantitatively explain splittings of atomic energy levels in an applied external magneticfield (the Zeeman effect). The same value is obtained from the non-relativistic reduction ofthe Dirac equation. The full Quantum Electrodynamics predicts that the value 2 gets cor-rections and the measured values of the electron and muon magnetic moments constitutevery important and most precise tests of this theory.

86

Secondly, splittings of the atomic energy levels due to relativistic effects(including the spin-orbit coupling) and interaction of the electron and nu-cleus spins (the hypefine splitting) can also be straightforwardly taken intoaccount. In the case of one-electron atoms this can be done rather preciselyeither by calculating (using the ordinary Rayleigh-Schrodinger perturbationtheory) corrections to the obtained from the Schrodinger or Pauli equationsenergy levels and/or the state-vectors or by using both elements obtained bysolving the relativistic Dirac wave equation.

Another major conceptual improvement of the theory of radiation (whichcan be imposed on the top of the improvements mentioned in the previ-ous paragraph because still the two sectors - matter and radiation - will betreated to a large extent separately) is the replacement of the charged parti-cle interaction with the classical continuous radiation field by its interactionwith the quantized electromagnetic radiation field represented by a collectionof quanta called photons. In this way one obtains a combined theory of twoquantum systems - of the nonrelativistic (i.e. moving with velocities v ≪ c)particle and the radiation field. This theory, while still incomplete (see theremarks at the end of this section), allows essentially to handle quantitativelymost of practical problems of interaction of radiation with matter.

The first step in the formulation of this theory is the quantization ofthe free electromagnetic field represented in the adopted radiation gauge,∇ ·A = 0, ϕ = 0, by the vector potential A, satisfying the wave equation(3.13). The electric and the magnetic fields are then giveny by E = −(1/c)A,B = ∇×A. If this field is enclosed in a box of a finite volume V = L3, theonly dynamical variables are the (complex) coefficients Ak of the expansions

A(r) =1√V

∑

k

(

Ak eik·r +A∗

k e−ik·r

)

, (3.124)

E(r) =i√V

∑

k

ωk

c

(

Ak eik·r −A∗

k e−ik·r

)

,

where ωk = c|k|. The expression for E(r) has been written down by ex-ploiting the fact that, as follows from the wave equation (3.13), Ak ∝ e−iωkt.In addition, the gauge condition ∇ ·A = 0 implies that k ·Ak = 0. Tomake contact with the quantization of ordinary systems classically charac-terized by real dynamical variables, we introduce two real combinations ofthe coefficients Ak

Qk =1√4π

(Ak +A∗k) , Pk = − iωk√

4π(Ak −A∗

k) , (3.125)

so that

Ak =√4π

1

2

(

Qk +i

ωk

Pk

)

, A∗k =

√4π

1

2

(

Qk −i

ωk

Pk

)

. (3.126)

87

The vector potential A(r) and the electric field E(r) expressed through thesereal variables take the forms

A(r) =

√

4π

V

∑

k

[

Qk cos(k·r)−1

ωk

Pk sin(k·r)]

, (3.127)

E(r) =

√

4π

V

∑

k

ωk

c

[

−Qk sin(k·r)−1

ωk

Pk cos(k·r)]

. (3.128)

Using now the integrals∫

V

d3r sin(k·r) sin(k′ ·r) = V

2(δk′,k − δ−k′,k) ,

∫

V

d3r sin(k·r) cos(k′ ·r) = 0 ,

etc., the energy∫

V

d3rw =1

8π

∫

V

d3r (E2 +B2) =1

8π

∫

V

d3r (E2 + ∂jA·∂jA) , (3.129)

of the electromagnetic field configuration represented by the vector potential(3.127) and its time derivative (times −1/c), that is the electric field E(r)(3.128) can be expressed through the variables Qk and Pk. (The last formof (3.129) follows upon exploiting the condition ∇ ·A = 0). The resultingexpression

∫

V

d3rw =1

2c2

∑

k

(P2k + ω2

kQ2k).

plays for the variables Qk and Pk the role of the Hamiltonian Hrad0 . Since

k · Pk = k · Qk = 0, it is convenient to introduce two (at this point) unitvectors ǫ(k, λ), λ = ±1 perpendicular to k and to write Pk =

∑

λ Pkλǫ(k, λ),Qk =

∑

λQkλǫ(k, λ), so that

Hrad0 =

1

2c2

∑

k,λ

(P 2kλ + ω2

kQ2kλ). (3.130)

This shows that the radiation field in the box of volume V behaves as a collec-tion of independent (uncoupled) harmonic oscillators and allows to identifyQkλ/c and Pkλ/c as two sets of canonically conjugated variables. Quantiza-tion of the radiation field then means promoting Qkλ and Pkλ to HermitianSchrodinger picture (i.e. time independent) operators Qkλ and Pkλ satisfyingthe commutation rules

[Qkλ, Pk′λ′ ] = i~c2δk′,kδλλ′ , (3.131)

[Qkλ, Qk′λ′] = [Pkλ, Pk′λ′] = 0 .

88

By analogy with the ordinary treatement of the harmonic oscillator one in-troduces the annihilation and creation operators

akλ =

√

ωk

2~c2

(

Qkλ +i

ωk

Pkλ

)

, a†kλ =

√

ωk

2~c2

(

Qkλ −i

ωk

Pkλ

)

,

satisfying the standard rules

[akλ a†k′λ′ ] = δk′,kδλλ′ , [akλ ak′λ′ ] = [a†kλ a

†k′λ′ ] = 0 . (3.132)

Expressed through the annihilation and creation operators the Schrodingerpicture operators A(r) and E(r) take the forms23

A(r) =√4π

∑

k,λ

√

~c2

2ωkV

[

eik·rakλǫ(k, λ) + e−ik·ra†kλǫ∗(k, λ)

]

, (3.133)

E(r) =√4π

∑

k,λ

iωk

c

√

~c2

2ωkV

[

eik·rakλǫ(k, λ)− e−ik·ra†kλǫ∗(k, λ)

]

. (3.134)

We have allowed here for complex vectors ǫ satisfying the conditions k ·ǫ = 0and ǫ

∗(k, λ) · ǫ(k, λ′) = δλλ′ . Since the Hamiltonian (3.130) takes the form

Hrad0 =

∑

k,λ

~ωk(a†kλakλ +

1

2) , (3.135)

its spectrum and its eigenvectors are determined by the standard algebraicargument (quoted in a footnote in section 1.3). Firstly, in the Hilbert spacethere is a vector |0, 0, . . .〉 (normalized to unity), which will be called herethe “photonic vacuum” and denoted |Ωγ

0〉, annihilated by all the operatorsakλ. It is the lowest energy eigenvector of Hrad

0 but, since the number ofindependent oscillators (to which the radiation field has been reduced) is(countably) infinite, the corresponding lowest Hrad

0 eigenvalue is also infinite.As the absolute values ofHrad

0 energy levels will not play any role in the theoryof radiation, it is most practical to declare simply that the true Hamiltonianis just

Hrad0 =

∑

k,λ

~ωka†kλakλ . (3.136)

23The awkward factor of√4π appears because we are using here - in order to keep

decorations from the heroic epoch - the Gauss system of units; this and the other factorsof ~, c and V are in the general approach to field quantization, discussed at lenght insection 11, fixed uniquely by the canonical commutation rules imposed on the operatorsA(r) and E(r); here we have arrived at the correct normalizations by making contact withthe canonical variables of a set of harmonic oscillators.

89

The other (normalizable) eigenvectors of Hrad0

|nk1λ1 , nk2λ2 , . . .〉 , (3.137)

have then finite energies Enk1λ1,nk2λ2

,... and are created by acting on |Ωγ0〉 with

the appropriate number of the creation operators. Together with |Ωγ0〉 the

vectors (3.137) span the “photonic” Hilbert space Hγ .

Transition to the infinite volume can be done with the help of the re-placements

∑

k

→ V

∫

d3k

(2π)3,

√V akλ → aλ(k) , (3.138)

so that in the continuum [aλ(k), a†λ′(k′)] = (2π)3δ(3)(k′ − k) δλλ′ and

Hrad0 =

∫

d3k

(2π)3~ωk

∑

λ

a†λ(k)aλ(k) . (3.139)

Of course in the continuum all the eigenvectors |k1λ1,k2λ2, . . .〉 of (3.139),except for the photonic vacuum |Ωγ

0〉, become generalized (i.e. nonnormaliz-able) eigenvectors.

One can now construct a quantum theory of the charged particle inter-action with the quantized radiation field by combining the free radiationHamiltonian (3.136) with some matter Hamiltonian Hmatt

0 (e.g. the Hamil-tonian (3.20) describing a single electron bound in the potential V (r)) andthe interaction term, e.g.24

Vint = − qe

McA(r)·p+

q2e2

2Mc2A2(r) . (3.140)

The Hilbert space H of such a theory is spanned by all tensor products of theHmatt

0 eigenvectors (normalizable) |Ωγ0〉 and (nonnormalizable in the contin-

uum) vectors (3.137) and the eigenvectors (normalizable and nonormalizable)of Hmatt

0 ; in the particular case of a single charged spinless particle

H = L2(R3)⊗Hγ .

It is interesting to reconsider the processes of absoption and emissionof light by atoms within this quantum theory of radiation. Probabilitiesper unit time of transitions between the atomic states |ai〉 and |af〉 (theeigenvectors of Hmatt

0 with the eigenvalues Eai and Ea

f , respectively) which

24Note that the spatial argument r of the operator (3.133) has got repaced here by theposition operator r acting in the Hilbert space spanned by the eigenvectors of Hmatt

0 .

90

are now accompanied by a corresponding change of the state of the radiationfield, are obtained using the Fermi’s Golden Rule (2.42) applied to the time-independent perturbation (3.140) from which we will, as previously, omit thesecond term quadratic in the vector potential operator A. We will assumethat the volume V (in which the radiation field is quantized) is so large,that there exist photon frequencies ωk arbitrarily close to any atomic energylevel difference |Ea

f − Eai |/~. We consider |i〉 = |ai〉 ⊗ | . . . , nkλ, . . .〉 and

|f〉 = |af〉 ⊗ | . . . , nkλ ∓ 1, . . .〉 eigenvectors of H0 = Hmatt0 + Hrad

0 as theinitial and final states (other numbers nklλl

are the same in the initial andfinal photonic states). The matrix element squared of Vint is

|〈f |Vint|i〉|2 =e2

M2c24π~c2

2ωkV

nkλ |ǫ(k, λ)·〈af |eik·rp|ai〉|2 (Eaf > Ea

i )

(nkλ + 1) |ǫ∗(k, λ)·〈af |e−ik·rp|ai〉|2 (Eaf < Ea

i )

The upper form of the matrix element corresponds to the absorption in theatomic transition of one photon and the lower one to the emission of onephoton. The corresponding first order transition coefficient a

(1)fi squared is

|a(1)fi (t)|2 =sin2(ωfit/2)

~2(ωfi/2)2|〈f |Vint|i〉|2,

where now ωfi = (Eaf − Ea

i ∓ ~ωk)/~. From the discussion of section 2.5 it

is clear that for t large enough, only the factors |a(1)fi (t)|2 corresponding to|k| = ωk/c close to |Ea

f − Eai |/~c are numerically nonnegligible.

Let us consider first the absorption processes. Similarly as in section2.5, to obtain the constant transition probability per unit time we considertransitions from a group of initial states of energies Ei (with respect to H0 =Hmatt

0 +Hrad0 ) close to the energy Ef of the final state |af〉⊗| . . . , nkλ−1, . . .〉

with a fixed total number of photons equal the number of the initial photonsminus one. Since the atomic energy levels are discrete, any variation of Ei

can only be realized by moving one photon in the initial and final statesbetween infinitesimally close “bins”, i.e. by considering transitions from thestates of the form (recall that Vint can change the number of photons onlyby one - we neglect the interaction term ∝ A2)

|ai〉 ⊗ | . . . , nkλ − 1, nk+dkλ + 1, . . .〉 ,

of energies infinitesimally different from the energy of the corresponding finalstate |af〉⊗ | . . . , nkλ−1, nk+dkλ, . . .〉.25 Thus, effectively the density ρ(Ei) of

25Since the photonic initial and final states must be correlated in this way, it is in factimmaterial whether we consider transition from a group of initial states or to a group offinal states. We have chosen to talk about a group of initial states to make closer theanalogy with the classical picture of radiation.

91

the initial states can be counted in the same way as the density of states ofa single photon and the number of initial states in the interval (Ei, Ei+ dEi)is equal

dEi ρ(Ei) =V

(2π)3d3k =

V

(2π)3ω2k

~c3dEidΩk (3.141)

Proceeding then exactly as in section 2.5, i.e. taking the matrix elementoutside the integral over dEi and then extending the integration to the wholereal axis, we arrive at the probability per unit time

w(i→ f) =2π

~

e2

M2c24π~c2

2ωkV|ǫ(k, λ)·〈af |eik·rp|ai〉|2 nkλ

V

(2π)3ω2k

~c3dΩk .

To make a contact with the caculation of wind(i→ f) done in section 3.2,it is necessary to identify the flux spectral density I(ω). This is possible onlyif the numbers of photons in “bins” corresponding to the considered groupof initial state are much larger than 1, so that they can make up a classicalelectromagnetic field. If this is so, in the interval dωkdΩk there are

V

(2π)3ω2k

c3dωkdΩk

types of photons (types of vibrations of the classical electromagnetic field)and each type contributes nkλ~ωk to the energy of the photons (of the electro-magnetic field) in the volume V . Hence, the energy density of the radiationdue to photons having the wave vector k in the solid angle dΩk (and polar-ization λ) and frequency in the interval dωk, contained in the volume V , thatis its spectral density ρ(ω, λ) should be identified with

ρ(ω, λ) =nkλ

(2π)3~ω3

k

c3dΩk

Expressing the transition probability w(i → f) through I(ω, λ) = cρ(ω, λ)

and recalling the definition (3.29) of the p(±)fi factors one recovers the formula

(3.31) for wind(i→ f)

wind(i→ f) =4π2

~2cI(ωfi, λ) |ǫ(k, λ)·p(+)

fi |2.

The modulus squared of the matrix element of Vint relevant for transitionsin which one photon is created is proportional to nkλ+1. In this formulationthere is in fact no distinction between induced and spontaneous emissions:one can consider transitions with an arbitrary number of photons in the

92

initial state (in particular, wit no photons at all) and one photon more inthe final state. Induced transitions can meaningfully be defined only if thenumber of photons (in the relevant frequency interval) is sufficiently large.In this case one can again make contact with the calculation of sections3.2 and 3.3 by treating the contributions proportional to nkλ and to “1” tothe probability of the transition differently. Applying the same procedureas above to the contribution proportional to nkλ one readily recovers theformula (3.32) proportional to I(|ωfi|, λ) for the probability per unit time ofthe induced transition in which the atom deexcites from the higher energystate |ai〉 to the lower state |af 〉.

Considering the contribution proportional to “1”, one computes insteadthe probability per unit time of the transition from a state with fixed numbersnkλ of photons and the atom in the state |ai〉 to a group of final states withone photon more in one of the “bins” corrsponding to the photon energiesin the range k and k + dk, where |k| = (Ea

i − Eaf )/~c. The counting of

these states is again equivalent to the counting of states of a single photon;therefore their number is given by (3.141) and from the Fermi’s Golden Rule(2.42) one obtains

dwspont(i→ f) =2π

~

e2

M2c24π~c2

2ωkV

∣

∣

ǫ∗(k, λ) ·〈af |e−ik·rp|ai〉

∣

∣

2 V

(2π)3ω2k

~c3dΩk .

In the dipole approximation

ǫ∗(k, λ) ·〈af |e−ik·rp|ai〉 ≈ iMǫ

∗(k, λ)·〈af |r|ai〉(Eaf − Ea

i )/~ ,

and (since (Eaf − Ea

i )/~ = ωk)

dw1E

spont(i→ f) ≈ 4π2e2 |ǫ∗(k, λ) ·〈af |r|ai〉|21

(2π)3ω3k

~c3dΩk .

Upon summing over the possible polarizations λ of the emitted photon usingthe formula (3.48) and integrating over its directions (with the help of (3.49))one recovers the formula (3.77).

Another application of this theory of radiation, showing that it has muchwider domain of applicability than the semiclassical one. is the calculationof the cross section of the Compton process, i.e. of the elastic scattering ofphotons on free electrons. Of course since the “matter part” of the theory isbased on the nonrelativistic quantum mechanics, the cross section obtained inthis way can only be valid in the nonrelativistic regime, i.e. when ~ω ≪ Mc2.Again, it is sufficient to consider the one-particle quantum mechanics basedon the Schrodinger equation (if the electron spin can be neglected), so that

93

the Hilbert space is H = L2(R3) ⊗ Hγ. Since the electron on which the

photon is scattered is not bound,

H0 =

∫

d3k

(2π)3~ωk

∑

λ=±1

a†λ(k) aλ(k) +p2

2M, (3.142)

and the interaction is

Vint =e

2Mc

(

p·A(r) + A(r)·p)

+e2

2Mc2A2(r) . (3.143)

Owing to the gauge condition ∇·A = 0, it can be simplified to

Vint =e

McA(r)·p+

e2

2Mc2A2(r) ≡ V

(1)int + V

(2)int . (3.144)

The operator A is given in terms of the creation and annihilation operatorsby (we use the normalization in the continuum here)

A(r) =√4π

∫

d3k

(2π)3

√

~c2

2ωk

∑

λ=±1

[

eik·raλ(k)ǫ(k, λ) + e−ik·ra†λ(k)ǫ∗(k, λ)

]

.

Considering the Compton process, we are interested in computing theprobability per unit time of the transition between the initial |i〉 and final|f〉 eigenvectors of H0 (3.142) representing the states of one electron and onephoton:

|i〉 = a†λ1(k1)|Ωγ

0〉 ⊗ |p1〉 ,|f〉 = a†λ2

(k2)|Ωγ0〉 ⊗ |p2〉 . (3.145)

The required transition probability per unit time wfi can be computed fromthe Fermi’s Golden Rule extended to higher orders

wfi =2π

~δ(Ef − Ei)

∣

∣

∣

∣

∣

〈f |Vint|i〉+∑

ℓ

〈f |Vint|ℓ〉〈ℓ|Vint|i〉Ei −Eℓ + i0

+ . . .

∣

∣

∣

∣

∣

2

, (3.146)

in which the sum over ℓ extends to all possible intermediate states. Since theinitial and final states (3.145) have the same numbers of photons, it is clear

that in the first term under the modulus squared only V(2)int can contribute,

while the nonzero contribution of V(1)int in the second term arises from the

intermediate states |ℓ〉 with zero or two photons. It is also clear that together

the contributions of V(2)int to the first term and of V

(1)int to the second one

give the first, order e2, contribution to the process amplitude and order e4,contribution to the transition probability.

94

Computation of 〈f |V (2)int |i〉 is straightforward: of the product of two op-

erators A a nonzero contribution arises only from the two terms with onecreation and one annihilation operator. This matrix element factorizes intothe product of the “photonic” matrix element

4πe2

2Mc2

∑

λ,λ′

∫

d3k

(2π)3d3k′

(2π)3

√

~c2

2ωk

√

~c2

2ωk′

ǫ(k, λ) ·ǫ∗(k′, λ′)

×〈Ωγ0 |aλ2(k2) [aλ(k)a

†λ′(k

′) + a†λ′(k′)aλ(k)] a

†λ1(k1)|Ωγ

0〉 ,

(in one of the terms the primed and unprimed labels have been interchanged)and the “electronic” one

〈p2|e−ik·r eik′·r|p1〉 =

∫

d3r ei(p1−p2)·rei(k−k′)·r = (2π)3δ(3)(k′ + p2 − k− p1) .

To evaluate the “photonic” matrix element it is convenient to interchangethe first pair of operators in the square bracket by writing aλ(k)a

†λ′(k′) =

a†λ′(k′)aλ(k)+(2π)3δ(3)(k−k′)δλ′λ. After this operation there are two identicalterms in addition to the one with the delta function. Acting now with theannihilation operators to the right and with the creation operators to theleft, one gets ultimately two c-number terms which can be combined nextwith the “electronic” matrix element. The integrals over d3k and d3k′ canbe then taken owing to the delta functions (similarly disappear the sumsover polarizations λ and λ′). The evaluation of the entire matrix element

〈f |V (2)int |i〉 can be represented graphically by the diagrams shown in figure

3.1. The contribution 〈f |V (2)int |i〉(1) associated with the first of the diagrams

indicated in this figure by the arrow to the matrix element 〈f |V (2)int |i〉 reads

〈f |V (2)int |i〉(1) =

e2~

2M(2π)3δ(3)(k2 + p2 − k1 − p1) 4π

2 ǫ∗(k2, λ2)·ǫ(k1, λ1)√2ωk2

√2ωk1

.

The contribution 〈f |V (2)int |i〉(2) of the last diagram of figure 3.1 involves a

divergent integral. It should be discarded (see the comments at the endof this section). Equivalently, one can maintain that just as the infiniteenergy of the Hrad

0 ground state has been (artificially) subtracted by normalordering the operators in Hrad

0 , also the correct interaction Vint should benormal ordered, that is the last term of (3.143) (and of (3.144)) should read(e2/2Mc2) : A(r)·A(r) :. The last diagram of figure 3.1 is then absent.

On the basis of the middle diagram of figure 3.1 the structure of theobtained matrix element of V

(2)int can be interpreted in the following way:

with each incoming (i.e. corresponding to the particle in the initial state)spin 0 particle dashed line (recall, that electron is treated here as spinless!)

95

p1

p2

k1λ1

k2λ2

×

×

×

×

−→

p1

p2

k1λ1

k2λ2

×

×

×

×

+

p1

p2

k1λ1

k2λ2

×

×

×

×

Figure 3.1: Graphical evaluation of the matrix element 〈f |V (2)int |i〉. The arrow

indicates that the set of graphical elements (two external photon lines, twoexternal electron lines and one interaction vertex) can be assembled in twodifferent ways resulting in two different diagrams which correspond to thetwo terms obtained by evaluating the matrix element 〈f |V (2)

int |i〉(2) using thecommutation rules of the photonic creation and annihilation operators.

and photon wavy line associated are the factors

1 , and√4π

ǫ(k1, λ1)√2ωk1

,

and with each outgoing (i.e. corresponding to the particle in the final state)electron and photon line associated are the factors

1 , and√4π

ǫ∗(k2, λ2)√2ωk2

.

With the interaction vertex associated is the factor

e2~

2M(2π)3δ(3)(k2 + p2 − k1 − p1) .

Since in combining the elements shown in the left part of figure 3.1 into thefirst diagram on the right there are two possibilities of attaching the twoexternal photon lines to the vertex, the combinatoric factor of 2 appears inthe contribution to the transition amplitude corresponding to this diagram.These associations are the prototypes of the Feynman rules which will besystematically introduced (in a somewhat different setting) in section 9.

To find the contribution to the transition amplitude of V(1)int one has to

consider intermediate states |ℓ0γ〉 with one electron and no photons and |ℓ2γ〉with one electron and two photons. In the contribution of |ℓ0γ〉 the product

of the matrix elements 〈f |V (1)int |ℓ0γ〉〈ℓ0γ |V (1)

int |i〉 in the formula (3.146) is

e2

M2c2

∫

d3q

(2π)3〈p2| ⊗ 〈Ωγ

0 |aλ2(k2) A(r)·p |Ωγ0〉 ⊗ |q〉

〈q| ⊗ 〈Ωγ0 | A(r)·pa†λ1

(k1)|Ωγ0〉 ⊗ |p1〉 (3.147)

96

After evaluation one gets for 〈f |V (1)int |ℓ0γ〉〈ℓ0γ|V (1)

int |i〉 the expression

e2~3

M2(2π)3δ(3)(k2 + p2 − k1 − p1) 4π

p2 ·ǫ∗(k2, λ2)√2ωk2

p1 ·ǫ(k1, λ1)√2ωk1

.

The energy denominator corresponding to the zero-photon intermediate statein the formula (3.146) is

Ei −Eℓ0γ = ~ωk1 +~2p2

1

2M− ~

2(p1 + k1)2

2M.

Again the whole expression obtained for 〈f |V (1)int |ℓ0γ〉〈ℓ0γ|V (1)

int |i〉/(Ei − Eℓ0γ )can be given the following interpretation (illustrated graphically in figure3.2). It is assembled in the way dictated by the first diagram after the arrowout of the following elements: in addition to the already identified factorscorresponding to the external electron and photon lines, the factor26

e~3/2

2M(p′ + p) (2π)3δ(3)(p′ − p∓ k) ,

is associated with the interaction vertex into which electron and photon withthe wave vector p and k enter (the upper minus sign in the Dirac delta)and electron with the wave vector p′ leaves. (If the photon is outgoing, thelower plus sign applies). The explicit wave vector factor p′ + p should getcontracted with the polarization vector of the external photon line attachedto this vertex.27 To the energy denominator corresponds the factor

~ωk1 +~2p2

1

2M− ~

2q2

2M,

and the whole expression (involving two three-dimensional Dirac deltas) isintegrated over d3q/(2π)3 (summation over all possible intermediate zero-photon states). This integration produces the Dirac delta expressing theoverall conservation of the wave vectors (three-momentum conservation) andsubstitutes p1 + k1 for q into the energy denominator.

With the rules just formulate it is straightforward to write down theexpression corresponding to the second diagram after the arrow in figure 3.2:it is given by

e2~3

M2(2π)3δ(3)(k2 + p2 − k1 − p1) 4π

p1 ·ǫ∗(k2, λ2)√2ωk2

p1 ·ǫ(k1, λ1)√2ωk1

.

26This is more evident if the interaction V(1)int is kept in the form as in the formula

(3.143).27As we will argue, there is no point to discuss here the possiblility that the photon line

is not external but comes from another interaction vertex.

97

p1

p2

k1λ1

k2λ2

×

×

×

×

−→

p1

p2

k1λ1

k2λ2

×

×

×

×

p1

p2

k1λ1

k2λ2

|ℓ0γ〉 +

×

×

×

×

p1

p2

k1λ1

k2λ2

|ℓ2γ〉 +

p1

p2

k1λ1

k2λ2

×

×

×

×

|ℓ2γ〉

Figure 3.2: Graphical interpretation of the contribution of second order inV

(1)int to the transition amplitude. Horizontal dotted lines mark the interme-

diate state |ℓ〉.

divided by the energy denominator

Ei − Eℓ2γ = ~ωk1 +~2p2

1

2M− ~

2(p1 − k2)2

2M− ~ωk1 − ~ωk2

=~2p2

1

2M− ~

2(p1 − k2)2

2M− ~ωk2 .

The last diagram of figure 3.2 involves a divergent integral and cannot begiven any sensible interpretation within the developed approach. It is also noteliminated by normal ordering of the Hamiltonian H = Hrad

0 +Hmatt0 + Vint.

We simply discard it.

The combined expressions corresponding to the diagrams of figures 3.1and 3.2 allow to find the amplitude of the transition between the initial andfinal states (3.145) for any kinematical configuration, that is for arbitraryp1 and k1 (respecting of course the limitation ~c|k1| ≪ Mc2). Here we willrestrict the calculation to the scattering of photons on free electrons at rest,by setting p1 = 0. This simplifies the task leaving ony the contribution ofV

(2)int . In using the formula (3.146) the square of the Dirac delta function to

which the matrix element 〈f |V (2)int |i〉 is proportional should be interpretted in

the following way

[(2π)3δ(3)(Pf −Pi)]2 =

∫

d3x ei(Pf−Pi)·x (2π)3δ(3)(Pf −Pi)

= V (2π)3δ(3)(Pf −Pi) .

As will be explained in section 10, the proper quantity corresponding to whatis measured experimentally is in this case not the transition probability perunit time, but rather the transition probability per unit time per unit volume.Dividing by the volume V we get

wfi =2π

~2δ(ωk2 + ~p2

2/2M − ωk1)

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×∣

∣

∣

∣

4πe2~

2M

2 ǫ∗(k2, λ2)·ǫ(k1, λ1)√2ωk2

√2ωk1

∣

∣

∣

∣

2

(2π)3δ(3)(k2 + p2 − k1) .

To obtain the scatterng cross section characterizing the probability of indi-vidual scattering events one has to consider transitions to a group of finalstates (see section 2.4), that is to multiply wfi by [d3k2/(2π)

3][d3p2/(2π)3]

and to divide by the relative flux of incoming particles. With the adoptednormalization of the free electron and photon states, which correspond to oneparticle in the unit volume,28 the latter step reduces to dividing the resultingexpression by c. The integral over d3p can be then immediately performedusing the three-dimensional Dirac delta function. In this way one arrives at(recall, e2/~c = αEM; we also simplify the notation writing ǫ1 for ǫ(k1, λ1),ω1 for ωk1, etc.)

dσ = 16π2 (~c)2

c

α2EM

4M2

|ǫ∗2 ·ǫ1|2ω2ω1

2πδ

(

ω2 − ω1 +~(k2 − k1)

2

2M

)

d3k2

(2π)3.

Writing now (k2 − k1)2 = (ω2

2 + ω21 − 2ω2ω1 cos θ)/c

2, where θ is the anglebetween k2 and k1, and d

3k2 as (ω22/c

3)dΩk2dω2 the integral over dω2 can beperformed exploiting the remaining delta function. This gives the formula

dσ = (~c)2(αEM

Mc2

)2 |ǫ∗2 ·ǫ1|2ω1

ω2

1 + (~/Mc2)(ω2 − ω1 cos θ)dΩk2 ,

in which ω2 is the root of the quadratic equation

ω22 + 2

(

Mc2

~− ω1 cos θ

)

+ ω21 −

2Mc2

~ω1 = 0 .

which, in the regime ~ω1 ≪Mc2, gives ω2 ≈ ω1[1−(~ω1/Mc2)(1−cos θ)+. . .].Therefore the final (“nonrelativistic”) formula for the differential Comptonscattering cross section is

dσ

dΩk2

= (~c)2(αEM

Mc2

)2

|ǫ∗2 ·ǫ1|21− (~ω1/Mc2)(1− cos θ)

1 + (~ω1/Mc2)(1− cos θ).

This agrees up to terms of order (~ω1/Mc2)0 with the relativistic Klein-Nishijima cross section of the Compton scattering on pointlike spin 1

2elec-

trons

dσ

dΩk2

= (~c)2( αEM

2Mc2

)2(

ω2

ω1

)2 [ω2

ω1+ω1

ω2+ 4|ǫ∗2 ·ǫ1|2 − 2

]

,

28Recall that we use 〈r|p〉 = eip·r; therefore |〈r|p〉|2 integrated over the unit volume isjust 1.

99

and up to terms of order (~ω1/Mc2)1 with the relativistic formula for theCompton scattering on pointlike spinless particles

dσ

dΩk2

= (~c)2(αEM

Mc2

)2 |ǫ∗2 ·ǫ1|2[1 + (~ω1/Mc2)(1− cos θ)]2

.

It should be stressed that theory developed here (and its variants with the“matter part” replaced by something more complicated) is not the full quan-tum electrodynamics because it has not been properly quantized (missedis for example the Coulombic interaction between charges). Also the con-structed theory is not relativistic. Antiparticles required by relativistic invari-ance (see section 8) have not been introduced into it. All this will be improvedin due course. Nevertheless, it can account for an extremely large amount ofdata related to radiation emited and/or absorbed by atoms, molecules andnuclei as well as scattering of light on these objects. Furthermore, changingthe basis in the photonic Hilbert space one can construct states represent-ing photons of definite total angular momentum and investigate in this waymultipole structure of the radiation emitted by nuclei for example (half ofthe book by Bierestecki, Lifszyc & Pitajewski is about this)

It is also important to remember that essential tool in all the calculationsdone in this chapter was the Fermi Golden Rule applied to compute theprobabilities of transitions between the eigenstates of the free Hamiltonian.As has been seen on the example of Compton scattering this leads to somecontributions which have no direct physical interpretation and have to besimply rejected. In the full electrodynamics (as in any quantum field theory)the way of posing the questions will have to change (form the Bohr’s times weaccept that it is the theory which dictates which questions can legitimatelybe asked). The proper formulation of the theory will lead us to study nottransitions not between the eigenstates of the free Hamiltonian but betweenthe special class of eigenstates of the complete Hamiltonian, the in and out

states analogous to the in and out states introduced in section 1.3. It willbe then necessary to recognize that an object like a free atom (or a freeparticle) which is the eigenstate of the free Hamiltonian cannot be realizedexperimentally. For this reason, for example, there will be no point to askabout corrections to the Einstein’s coefficients defined in section 3.5. Withthe proper formulation however the cotributions like the ones which had tobe discarded in the computation of the Compton process cross section willbe handled in a logically consistent way.

100