[N (V ) , N (V )] = . (3.209)

|r ’ r |

4π

V V

A straightforward estimate shows that [N (V ) , N (V )] ∼ R’3 . Thus the commutator

between these proposed local number operators does not vanish for nonoverlapping

volumes; indeed, it does not even decay very rapidly as the separation between the

volumes increases. This counterintuitive behavior is caused by the nonlocal ¬eld com-

mutator (3.16) which is a consequence of the transverse nature of the electromagnetic

¬eld.

The alternative de¬nition (Deutsch and Garrison, 1991a),

20

d3 rE(’) (r) · E(+) (r) ,

G (V ) = (3.210)

ω0 V

of a local number operator is suggested by the Glauber theory of photon detection,

which is discussed in Section 9.1.2. Rather than anticipating later results we will obtain

½¼

Exercises

eqn (3.210) by a simple plausibility argument. The representation (3.39) for the ¬eld

Hamiltonian suggests interpreting 2 0 E(’) · E(+) as the energy density operator. For a

monochromatic ¬eld state this in turn suggests that 2 0 E(’) · E(+) / ω0 be interpreted

as the photon density operator. The expression (3.210) is an immediate consequence

of these assumptions. The integrand in this equation is clearly positive de¬nite, but

nonlocal e¬ects show up here as well.

The failure of several plausible candidates for a local number operator strongly

suggests that there is no such object. If this conclusion is supported by future research,

it would mean that photons are nonlocalizable in a very fundamental way.

3.7 Exercises

3.1 The ¬eld commutator

Verify the expansions (2.101) and (2.103), and use them to derive eqns (3.1) and (3.3).

3.2 Uncertainty relations for E and B

(1) Derive eqn (3.4) from eqn (3.3).

(2) Consider smooth distributions of classical polarization P (r) and magnetization

M (r) which vanish outside ¬nite volumes VP and VM respectively, as in Section

2.5. The interaction energies are

WE = ’ d3 rP (r) · E (r) , WB = ’ d3 rM (r) · B (r) .

Show that

i

[WB , WE ] = ’ d3 rP (r) · M (r) .

0

(3) What assumption about the volumes VP and VM will guarantee that WB and WE

are simultaneously measurable?

(4) Use the standard argument from quantum mechanics (Bransden and Joachain,

1989, Sec. 5.4) to show that WB and WE satisfy an uncertainty relation

∆WB ∆WE K,

and evaluate the constant K.

3.3 Electromagnetic Hamiltonian

Carry out the derivation of eqns (3.37)“(3.41).

3.4 Electromagnetic momentum

Fill in the steps leading from the classical expression (3.42) to the quantum form (3.48)

for the electromagnetic momentum operator.

Milonni™s quantization scheme—

3.5

Fill in the details required to go from eqn (3.159) to eqn (3.164).

½½¼ Field quantization

Electromagnetic angular momentum—

3.6

Carry out the calculations needed to derive eqns (3.172)“(3.178).

Wave packet quantization—

3.7

(1) Derive eqns (3.192), (3.193), and (3.195).

(2) Derive the expression for 1w |1v , where w and v are wave packets in “em .

4

Interaction of light with matter

In the previous chapters we have dealt with the free electromagnetic ¬eld, undisturbed

by the presence of charges. This is an important part of the story, but all experiments

involve the interaction of light with matter containing ¬nite amounts of quantized

charge, e.g. electrons in atoms or conduction electrons in semiconductors. It is there-

fore time to construct a uni¬ed picture in which both light and matter are treated by

quantum theory. We begin in Section 4.1 with a brief review of semiclassical electrody-

namics, the standard quantum theory of nonrelativistic charged particles interacting

with a classical electromagnetic ¬eld. The next step is to treat both charges and ¬elds

by quantum theory. For this purpose, we need a Hilbert space describing both the

charged particles and the quantized electromagnetic ¬eld. The necessary machinery is

constructed in Section 4.2. We present the Heisenberg-picture description of the full

theory in Sections 4.3“4.7. In Sections 4.8 and 4.9, the interaction picture is introduced

and applied to atom“photon coupling.

4.1 Semiclassical electrodynamics

In order to have something reasonably concrete to discuss, we will consider a system of

N point charges. The pure states are customarily described by N -body wave functions,

ψ (r1 , . . . , rN ), in con¬guration space. The position and momentum operators rn and

pn for the nth particle are respectively de¬ned by

rn ψ (r1 , . . . , rN ) = rn ψ (r1 , . . . , rN ) ,

(4.1)

‚

pn ψ (r1 , . . . , rN ) = ’i ψ (r1 , . . . , rN ) .

‚rn

The Hilbert space, Hchg , for the charges consists of the normalizable N -body wave

functions, i.e.

d3 rN |ψ (r1 , . . . , rN )|2 < ∞ .

d3 r1 · · · (4.2)

In all applications some of the particles will be fermions, e.g. electrons, and others will

be bosons, so the wave functions must be antisymmetrized or symmetrized accordingly,

as explained in Section 6.5.1.

In the semiclassical approximation the Hamiltonian for a system of charged parti-

cles coupled to a classical ¬eld is constructed by combining the correspondence prin-

ciple with the idea of minimal coupling explained in Appendix C.6. The result is

N N

2

(pn ’ qn A (rn , t))

Hsc = + qn • (rn , t) , (4.3)

2Mn

n=1 n=1

½½¾ Interaction of light with matter

where A and • are respectively the (c-number) vector and scalar potentials, and qn

and Mn are respectively the charge and mass of the nth particle. In this formulation

there are two forms of momentum: the canonical momentum,

‚

pn,can = pn = ’i , (4.4)

‚rn

and the kinetic momentum,

pn,kin = pn ’ qn A (rn , t) . (4.5)

The canonical momentum is the generator of spatial translations, while the classical

momentum M v is the correspondence-principle limit of the kinetic momentum.

It is worthwhile to pause for a moment to consider where this argument has led

us. The classical ¬elds A (r, t) and • (r, t) are by de¬nition c-number functions of

position r in space, but (4.3) requires that they be evaluated at the position of a

charged particle, which is described by the operator rn . What, then, is the meaning of

A (rn , t)? To get a concrete feeling for this question, let us recall that the classical ¬eld

can be expanded in plane waves exp (ik · r ’ iωk t). The operator exp (ik · rn ) arising

from the replacement of rn by rn is de¬ned by the rule

eik·rn ψ (r1 , . . . , rN ) = eik·rn ψ (r1 , . . . , rN ) , (4.6)

where ψ (r1 , . . . , rN ) is any position-space wave function for the charged particles.

In this way A (rn , t) becomes an operator acting on the state vector of the charged

particles. This implies, for example, that A (rn , t) does not commute with pn , but

instead satis¬es

‚Ai

[Ai (rn , t) , pnj ] = i (rn , t) . (4.7)

‚rj

The scalar potential • (rn , t) is interpreted in the same way.

The standard wave function description of the charged particles is useful for deriv-

ing the semiclassical Hamiltonian, but it is not particularly convenient for the applica-

tions to follow. In general it is better to use Dirac™s presentation of quantum theory, in

which the state is represented by a ket vector |ψ . For the system of charged particles

the two versions are related by

ψ (r1 , . . . , rN ) = r1 , . . . , rN |ψ , (4.8)

where |r1 , . . . , rN is a simultaneous eigenket of the position operators rn , i.e.

rn |r1 , . . . , rN = rn |r1 , . . . , rN , n = 1, . . . , N . (4.9)

In this formulation the wave function ψ (r1 , . . . , rN ) simply gives the components of

the vector |ψ with respect to the basis provided by the eigenvectors |r1 , . . . , rN . Any

other set of basis vectors for Hchg would do equally well.

½½¿

Quantum electrodynamics

4.2 Quantum electrodynamics

In semiclassical electrodynamics the state of the physical system is completely de-

scribed by a many-body wave function belonging to the Hilbert space Hchg de¬ned by

eqn (4.2), but this description is not adequate when the electromagnetic ¬eld is also

treated by quantum theory. In Section 4.2.1 we show how to combine the charged-

particle space Hchg with the Fock space HF , de¬ned by eqn (3.35), to get the state

space, HQED , for the composite system of the charges and the quantized electromag-

netic ¬eld. In Section 4.2.2 we construct the Hamiltonian for the composite charge-¬eld

system by appealing to the correspondence principle for the quantized electromagnetic

¬eld.

4.2.1 The Hilbert space

In quantum mechanics, many-body wave functions are constructed from single-particle

wave functions by forming linear combinations of product wave functions. For example,

the two-particle wave functions for distinguishable particles A and B have the general

form

ψ (rA , rB ) = C1 ψ1 (rA ) χ1 (rB ) + C2 ψ2 (rA ) χ2 (rB ) + · · · . (4.10)

Since wave functions are meaningless for photons, it is not immediately clear how

this procedure can be applied to the radiation ¬eld. The way around this apparent

di¬culty begins with the reminder that the wave function for a particle, e.g. ψ1 (rA ),

is a probability amplitude for the outcomes of measurements of position. In the stan-

dard approach to the quantum measurement problem”reviewed in Appendix C.2”a

measurement of the position operator rA always results in one of the eigenvalues rA ,

and the particle is left in the corresponding eigenstate |rA . If the particle is initially

prepared in the state |ψ1 A , then the wave function is simply the probability ampli-

tude for this outcome: ψ1 (rA ) = rA |ψ1 . The next step is to realize that the position

operators rA do not play a privileged role, even for particles. The components xA , yA ,

and zA of rA can be replaced by any set of commuting observables OA1 , OA2, , OA3

with the property that the common eigenvector, de¬ned by

OAn |OA1 , OA2 , OA3 = OAn |OA1 , OA2 , OA3 (n = 1, 2, 3) , (4.11)

is uniquely de¬ned (up to an overall phase). In other words, the observables OA1 , OA2 ,

OA3 can be measured simultaneously, and the system is left in a unique state after the

measurement.

With these ideas in mind, we can describe the composite system of N charges

and the electromagnetic ¬eld by relying directly on the Born interpretation and the

superposition principle. For the system of N charged particles described by Hchg , we

choose an observable O”more precisely, a set of commuting observables”with the

property that the eigenvalues Oq are nondegenerate and labeled by a discrete index q.

The result of a measurement of O is one of the eigenvalues Oq , and the system is left

in the corresponding eigenstate |Oq ∈ Hchg after the measurement. If the charges are

prepared in the state |ψ ∈ Hchg , then the probability amplitude that a measurement

½½ Interaction of light with matter

of O results in the particular eigenvalue Oq is Oq |ψ . Furthermore, the eigenvectors

|Oq provide a basis for Hchg ; consequently, |ψ can be expressed as

|ψ = |Oq Oq |ψ . (4.12)

q

In other words, the state |ψ is completely determined by the set of probability am-

plitudes { Oq |ψ } for all possible outcomes of a measurement of O.

The same kind of argument works for the electromagnetic ¬eld. We use box quanti-

zation to get a set of discrete mode labels k,s and consider the set of number operators

{Nks }. A simultaneous measurement of all the number operators yields a set of oc-

cupation numbers n = {nks } and leaves the ¬eld in the number state |n . If the ¬eld

is prepared in the state |¦ ∈ HF , then the probability amplitude for this outcome

is n |¦ . Since the number states form a basis for HF , the state vector |¦ can be

expressed as

|¦ = |n n |¦ ; (4.13)

n

consequently, |¦ is completely speci¬ed by the set of probability amplitudes { n |¦ }

for all outcomes of the measurements of the mode number operators. We have used

the number operators for convenience in this discussion, but it should be understood

that these observables also do not hold a privileged position. Any family of compatible

observables such that their simultaneous measurement leaves the ¬eld in a unique state

would do equally well.

The charged particles and the ¬eld are kinematically independent, so the operators

O and Nks commute. In experimental terms, this means that simultaneous measure-

ments of the observables O and Nks are possible. If the charges and the ¬eld are

prepared in the states |ψ and |¦ respectively, then the probability for the joint out-

come (Oq , n) is the product of the individual probabilities. Since overall phase factors

are irrelevant in quantum theory, we may assume that the probability amplitude for

the joint outcome”which we denote by Oq , n |ψ, ¦ ”is given by the product of the

individual amplitudes:

Oq , n |ψ, ¦ = Oq |ψ n |¦ . (4.14)

According to the Born interpretation, the set of probability amplitudes de¬ned by

letting Oq and n range over all possible values de¬nes a state of the composite system,

denoted by |ψ, ¦ . The vector corresponding to this state is called a product vector,

and it is usually written as

|ψ, ¦ = |ψ |¦ , (4.15)

where the notation is intended to remind us of the familiar product wave functions in

eqn (4.10).

The product vectors do not provide a complete description of the composite system,

since the full set of states must satisfy the superposition principle. This means that

we are required to give a physical interpretation for superpositions,

|Ψ = C1 |ψ1 , ¦1 + C2 |ψ2 , ¦2 , (4.16)

½½

Quantum electrodynamics

of distinct product vectors. Once again the Born interpretation guides us to the follow-

ing statement: the superposition |Ψ is the state de¬ned by the probability amplitudes

Oq , n |Ψ = C1 Oq , n |ψ1 , ¦1 + C2 Oq , n |ψ2 , ¦2

= C1 Oq |ψ1 n |¦1 + C2 Oq |ψ2 n |¦2 . (4.17)

It is important to note that for product vectors like |ψ |¦ the subsystems are

each described by a unique state in the respective Hilbert space. The situation is

quite di¬erent for superpositions like |Ψ ; it is impossible to associate a given state

with either of the subsystems. In particular, it is not possible to say whether the ¬eld

is described by |¦1 or |¦2 . This feature”which is imposed by the superposition

principle”is called entanglement, and its consequences will be extensively studied in

Chapter 6.

Combining this understanding of superposition with the completeness of the states

|Oq and |n F in their respective Hilbert spaces leads to the following de¬nition: the

state space, HQED , of the charge-¬eld system consists of all superpositions

|Ψ = Ψqn |Oq |n . (4.18)

q n

This de¬nition guarantees the satisfaction of the superposition principle, but the Born

interpretation also requires a de¬nition of the inner product for states in HQED . To this

end, we ¬rst take eqn (4.14) as the de¬nition of the inner product of the vectors |Oq , n

and |ψ, ¦ . Applying this de¬nition to the special choice |ψ, ¦ = |Oq , n yields

Oq , n |Oq , n = Oq |Oq n |n = δq ,q δn ,n , (4.19)

and the bilinear nature of the inner product ¬nally produces the general de¬nition:

¦— Ψqn .

¦ |Ψ = (4.20)

qn

q n

The description of HQED in terms of superpositions of product vectors imposes a

similar structure for operators acting on HQED . An operator C that acts only on the

particle degrees of freedom, i.e. on Hchg , is de¬ned as an operator on HQED by

C |Ψ = Ψqn {C |Oq } |n , (4.21)

q n

and an operator acting only on the ¬eld degrees of freedom, e.g. aks , is extended to

HQED by

aks |Ψ = Ψqn |Oq {aks |n } . (4.22)

q n

Combining these de¬nitions gives the rule

Caks |Ψ = Ψqn {C |Oq } {aks |n } . (4.23)

q n

½½ Interaction of light with matter

A general operator Z acting on HQED can always be expressed as

Z= Cn Fn , (4.24)

n

where Cn acts on Hchg and Fn acts on HF .

The o¬cially approved mathematical language for this construction is that HQED

is the tensor product of Hchg and HF . The standard notation for this is

HQED = Hchg — HF , (4.25)

and the corresponding notation |ψ —|¦ is often used for the product vectors. Similarly

the operator product Caks is often written as C — aks .

4.2.2 The Hamiltonian

For the ¬nal step to the full quantum theory, we once more call on the correspon-

dence principle to justify replacing the classical ¬eld A (r, t) in eqn (4.3) by the time-

independent, Schr¨dinger-picture quantum ¬eld A (r). The evaluation of A (r) at rn

o

is understood in the same way as for the classical ¬eld A (rn , t), e.g. by using the

plane-wave expansion (3.68) to get

eks aks eik·rn + HC .

A (rn ) = (4.26)

2 0 ωk V

ks

Thus A (rn ) is a hybrid operator that acts on the electromagnetic degrees of freedom

(HF ) through the creation and annihilation operators a† and aks and on the particle

ks

degrees of freedom (Hchg ) through the operators exp (±ik · rn ).

With this understanding we ¬rst use the identity

A (rn ) · pn + pn · A (rn ) = 2A (rn ) · pn ’ [Aj (rn ) , pnj ]

= 2A (rn ) · pn + i ∇ · A (rn ) , (4.27)

together with ∇ · A = 0 and the identi¬cation of • as the instantaneous Coulomb

potential ¦, to evaluate the interaction terms in the radiation gauge. The total Hamil-

tonian is obtained by adding the zeroth-order Hamiltonian Hem + Hchg to get

N N

2

(pn ’ qn A (rn ))

H = Hem + Hchg + + qn ¦ (rn ) . (4.28)

2Mn

n=1 n=1

Writing out the various terms leads to the expression

H = Hem + Hchg + Hint , (4.29)

1

: E2 : +µ’1 : B2 : ,

d3 r

Hem = (4.30)

0 0

2

½½

Quantum Maxwell™s equations

N

p2 1 qn ql

n

Hchg = + , (4.31)

|rn ’ rl |

2Mn 4π 0

n=1 n=l

N N 2

2

qn qn : A (rn ) :

=’ A (rn ) · pn +

Hint . (4.32)

Mn 2Mn

n=1 n=1

In this formulation, Hem is the Hamiltonian for the free (transverse) electromagnetic

¬eld, and Hchg is the Hamiltonian for the charged particles, including their mutual

Coulomb interactions. The remaining term, Hint , describes the interaction between

the transverse (radiative) ¬eld and the charges. As in Section 2.2, we have replaced

the operators E2 , B2 , and A2 in Hem and Hint by their normal-ordered forms, in order

to eliminate divergent vacuum ¬‚uctuation terms. The Coulomb interactions between

the charges”say in an atom”are typically much stronger than the interaction with

the transverse ¬eld modes, so Hint can often be treated as a weak perturbation.

4.3 Quantum Maxwell™s equations

In Section 4.2 the interaction between the radiation ¬eld and charged particles was

described in the Schr¨dinger picture, but some features are more easily understood

o

in the Heisenberg picture. Since the Hamiltonian has the same form in both pictures,