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d3 r
[N (V ) , N (V )] = . (3.209)
|r ’ r |

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,

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