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so ρ is a positive-de¬nite operator. Combining this with the normalization condition
Ψ |ρ| Ψ 1 for any normalized state |Ψ . The Born interpretation
(2.120) implies 0
2
tells us that | Ψe |Ψ | is the probability that a measurement”say of the projection
operator |Ψ Ψ|”will leave the system in the state |Ψ , given that the system is
prepared in the pure state |Ψe ; therefore, eqn (2.133) tells us that Ψ |ρ| Ψ is the
probability that a measurement will lead to |Ψ , if the system is described by the
mixed state with density operator ρ.
In view of the importance of the superposition principle for pure states, it is natural
to ask if any similar principle applies to mixed states. The ¬rst thing to note is that
linear combinations of density operators are not generally physically acceptable density
operators. Thus if ρ1 and ρ2 are density operators, the combination ρ = Cρ1 +Dρ2 will
be hermitian only if C and D are both real. The condition Tr ρ = 1 further requires
C + D = 1. Finally, the positivity condition (2.133) must hold for all choices of |Ψ ,
and this can only be guaranteed by imposing C 0 and D 0. Therefore, only the
convex linear combinations

ρ = Cρ1 + (1 ’ C) ρ2 , 0 C 1 (2.134)

are guaranteed to be density matrices. This terminology is derived from the mathe-
matical notion of a convex set in the plane, i.e. a set that contains every straight line
joining any two of its points. The general form of eqn (2.134) is

ρ= Cn ρn , (2.135)
n

where each ρn is a density operator, and the coe¬cients satisfy the convexity condition

0 Cn 1 for all n and Cn = 1 . (2.136)
n
Quantization of cavity modes

The o¬-diagonal matrix elements of the density operator are also constrained by the
de¬nition (2.116). The normalization of the ensemble states |Ψe implies | Ψe |Ψ | 1,
so

| Ψ |ρ| ¦ | = Pe Ψ |Ψe Ψe |¦
e

Pe | Ψ |Ψe | | Ψe |¦ | 1, (2.137)
e

i.e. ρ is a bounded operator.
The arguments leading from the ensemble de¬nition of the density operator to its
properties can be reversed to yield the following statement. An operator ρ that is
(a) hermitian, (b) bounded, (c) positive, and (d) has unit trace is a possible density
operator. The associated ensemble can be de¬ned as the set of normalized eigenstates
of ρ corresponding to nonzero eigenvalues. Since every density operator has a complete
orthonormal set of eigenvectors, this last remark implies that it is always possible to
choose the ensemble to consist of mutually orthogonal states.

2.3.4 Degrees of mixing
So far the distinction between pure and mixed states is absolute, but ¬ner distinc-
tions are also useful. In other words, some states are more mixed than others. The
distinctions we will discuss arise most frequently for physical systems described by a
¬nite-dimensional Hilbert space, or equivalently, ensembles containing a ¬nite number
of pure states. This allows us to simplify the analysis by assuming that the Hilbert
space has dimension d < ∞. The inequality (2.124) suggests that the purity
P (ρ) = Tr ρ2 1 (2.138)
may be a useful measure of the degree of mixing associated with a density operator ρ.
By virtue of eqn (2.122), P (ρ) = 1 for a pure state; therefore, it is natural to say that
the state ρ2 is less pure (more mixed) than the state ρ1 if P (ρ2 ) < P (ρ1 ). Thus the
minimally pure (maximally mixed) state for an ensemble will be the one that achieves
the lower bound of P (ρ). In general the density operator can have the eigenvalue
0 with degeneracy (multiplicity) d0 < d, so the number of orthogonal states in the
ensemble is N = d ’ d0 . Using the eigenstates of ρ to evaluate the trace yields
N
P (ρ) = p2 , (2.139)
n
n=1

where pn is the nth eigenvalue of ρ. In this notation, the trace condition (2.120) is just
N
pn = 1 , (2.140)
n=1

and the lower bound is found by minimizing P (ρ) subject to the constraint (2.140).
This can be done in several ways, e.g. by the method of Lagrange multipliers, with
the result that the maximally mixed state is de¬ned by
Mixed states of the electromagnetic ¬eld


, n = 1, . . . , N ,
1
N
pn = (2.141)
0 , n = N + 1, . . . , d .
In other words, the pure states in the ensemble de¬ning the maximally mixed state
occur with equal probability, and the purity is
1
P (ρ) =
. (2.142)
N
Another useful measure of the degree of mixing is provided by the von Neumann
entropy, which is de¬ned in general by
S (ρ) = ’ Tr (ρ ln ρ) . (2.143)
In the special case considered above, the von Neumann entropy is given by
N
S (ρ) = ’ pn ln pn , (2.144)
n=1

and maximizing this”subject to the constraint (2.140)”leads to the same de¬nition
of the maximally mixed state, with the value
S (ρ) = ln N (2.145)
of S (ρ). The von Neumann entropy plays an important role in the study of entangled
states in Chapter 6.

2.4 Mixed states of the electromagnetic ¬eld
2.4.1 Polarized light
As a concrete example of a mixed state, consider an experiment in which light from a
single atom is sent through a series of collimating pinholes. In each atomic transition,
exactly one photon with frequency ω = ∆E/ is emitted, where ∆E is the energy
di¬erence between the atomic states. The alignment of the pinholes determines the
unit vector k along the direction of propagation, so the experimental arrangement
determines the wavevector k = (ω/c) k. If the pinholes are perfectly circular, the
experimental preparation gives no information on the polarization of the transmitted
light. This means that the light observed on the far side of the collimator could be
described by either of the states
|Ψs = |1ks = a† |0 , (2.146)
ks

where s = ±1 labels right- and left-circularly-polarized light. Thus the relevant en-
semble is composed of the states |1k+ and |1k’ , with probabilities P+ and P’ , and
the density operator is
P+ 0
Ps |1ks 1ks | =
ρ= . (2.147)
P’
0
s

In the absence of any additional information equal probabilities are assigned to the
two polarizations, i.e. P+ = P’ = 1/2, and the light is said to be unpolarized. The
Quantization of cavity modes

opposite extreme occurs when the polarization is known with certainty, for example
P+ = 1, P’ = 0. This can be accomplished by inserting a polarization ¬lter after
the collimator. In this case, the light is said to be polarized, and the density operator
represents the pure state |1k+ . For the intermediate cases, a measure of the degree of
polarization is given by
P = |P+ ’ P’ | , (2.148)
which satis¬es 0 P 1, and has the values P = 0 for unpolarized light and P = 1
for polarized light.

A The second-order coherence matrix
The conclusions reached for the special case discussed above are also valid in a more
general setting (Mandel and Wolf, 1995, Sec. 6.2). We present here a simpli¬ed version
of the general discussion by de¬ning the second-order coherence matrix

Jss = Tr ρa† aks , (2.149)
ks

where the density operator ρ describes a monochromatic state, i.e. each state vector
|Ψe in the ensemble de¬ning ρ satis¬es ak s |Ψe = 0 for k = k. In this case we may
as well choose the z-axis along k, and set s = x, y, corresponding to linear polarization
vectors along the x- and y-axes respectively. The 2 — 2 matrix J is hermitian and posi-
tive de¬nite”see Appendix A.3.4”so the eigenvectors cp = (cpx , cpy ) and eigenvalues
np (p = 1, 2) de¬ned by
Jcp = np cp (2.150)
satisfy
c† cp = δpp and np 0. (2.151)
p
The eigenvectors of J de¬ne eigenpolarization vectors,
e p = c — e x + c— e y , (2.152)
px py

and corresponding creation and annihilation operators
a† = c— a† + c— a† , ap = cpx akx + cpy aky . (2.153)
px kx py ky
p

It is not di¬cult to show that
np = Tr ρa† ap , (2.154)
p

i.e. the eigenvalue np is the average number of photons with eigenpolarization ep . If ρ
describes an unpolarized state, then di¬erent polarizations must be uncorrelated and
the number of photons in either polarization must be equal, i.e.
n1 0
J= , (2.155)
20 1
where
n = Tr ρ a† akx + a† aky (2.156)
kx ky

is the average total number of photons. If ρ describes complete polarization, then
the occupation number for one of the eigenpolarizations must vanish, e.g. n2 = 0.
Mixed states of the electromagnetic ¬eld

Since det J = n1 n2 , this means that completely polarized states are characterized by
det J = 0. In this general setting, the degree of polarization is de¬ned by
|n1 ’ n2 |
P= , (2.157)
n1 + n2
where P = 0 and P = 1 respectively correspond to unpolarized and completely polar-
ized light.

B The Stokes parameters
Since J is a 2 — 2 matrix, we can exploit the well known fact”see Appendix C.3.1”
that any 2 — 2 matrix can be expressed as a linear combination of the Pauli matrices.
For this application, we write the expansion as
1 1 1 1
S0 σ0 + S1 σ3 + S2 σ1 ’ S3 σ2 ,
J= (2.158)
2 2 2 2
where σ0 is the 2 — 2 identity matrix and σ1 , σ2 , and σ3 are the Pauli matrices given
by the standard representation (C.30). This awkward formulation guarantees that the
c-number coe¬cients Sµ are the traditional Stokes parameters. According to eqn
(C.40) they are given by

S0 = Tr (Jσ0 ) , S1 = Tr (Jσ3 ) , S2 = Tr (Jσ1 ) , S3 = ’ Tr (Jσ2 ) . (2.159)

The Stokes parameters yield a useful geometrical picture of the coherence matrix, since
the necessary condition det (J) 0 translates to
2 2 2 2
S 1 + S2 + S3 S0 . (2.160)

If we interpret (S1 , S2 , S3 ) as a point in a three-dimensional space, then for a ¬xed
value of S0 the states of the ¬eld occupy a sphere”called the Poincar´ sphere”of
e
radius S0 . The origin, S1 = S2 = S3 = 0, corresponds to unpolarized light, since this
is the only case for which J is proportional to the identity. The condition det J = 0
for completely polarized light is
2 2 2 2
S 1 + S2 + S3 = S0 , (2.161)

which describes points on the surface of the sphere. Intermediate states of polarization
correspond to points in the interior of the sphere.
The Poincar´ sphere is often used to describe the pure states of a single photon,
e
e.g.
Cs a† |0 .
|ψ = (2.162)
ks
s

In this case S0 = 1, and the points on the surface of the Poincar´ sphere can be labeled
e
by the standard spherical coordinates (θ, φ). The north pole, θ = 0, and the south pole,
θ = π, respectively describe right- and left-circular polarizations. Linear polarizations
are represented by points on the equator, and elliptical polarizations by points in the
northern and southern hemispheres.
Quantization of cavity modes

2.4.2 Thermal light
A very important example of a mixed state arises when the ¬eld is treated as a thermo-
dynamic system in contact with a thermal reservoir at temperature T , e.g. the walls
of the cavity. Under these circumstances, any complete set of states can be chosen
for the ensemble, since we have no information that allows the exclusion of any pure
state of the ¬eld. Exchange of energy with the walls is the mechanism for attaining
thermal equilibrium, so it is natural to use the energy eigenstates”i.e. the number
states |n ”for this purpose.
The general rules of statistical mechanics (Chandler, 1987, Sec. 3.7) tell us
that the probability for a given energy E is proportional to exp (’βE), where
β = 1/kB T and kB is Boltzmann™s constant. Thus the probability distribution is
Pn = Z ’1 exp ’βEn , where Z ’1 is the normalization constant required to satisfy
eqn (2.112), and
En = ω κ nκ . (2.163)
κ

Substituting this probability distribution into eqn (2.116) gives the density operator

1 1
e’βEn |n n| =
ρ= exp (’βHem ) . (2.164)
Z Z
n

The normalization constant Z, which is called the partition function, is determined
by imposing Tr (ρ) = 1 to get

Z = Tr [exp (’βHem )] . (2.165)

Evaluating the trace in the number-state basis yields


exp ’β
Z= nκ ω κ = Zκ , (2.166)
n κ κ

where
∞ ∞
1
ωκ nκ
’βnκ ωκ
e’β
Zκ = e = = (2.167)
1 ’ e’β ωκ
nκ =0 nκ =0

is the partition function for mode κ (Chandler, 1987, Chap. 4).

A The Planck distribution
The average energy in the electromagnetic ¬eld is related to the partition function by

ωκ

’ ln Z = . (2.168)
U=
e β ωκ ’ 1
‚β κ

We will say that the cavity is large if the energy spacing c∆kκ between adjacent
discrete modes is small compared to any physically relevant energy. In this limit the
shape of the cavity is not important, so we may suppose that it is cubical, with
Mixed states of the electromagnetic ¬eld

κ ’ (k, s), where s = 1, 2 and ωκ ’ ck. In the limit of in¬nite volume, applying the
rule
d3 k
1
’ (2.169)
3
V (2π)
k
replaces eqn (2.168) by
U 2 ck
d3 k β ck
= . (2.170)
’1
3
V e
(2π)
After carrying out the angular integrations and changing the remaining integration
variable to ω = ck, this becomes

U
= dω ρ (ω, T ) , (2.171)
V 0

where the energy density ρ (ω, T ) dω in the frequency interval ω to ω + dω is given by
the Planck function
ω3
1
ρ (ω, T ) = 2 3 β ω . (2.172)
’1
πc e
B Distribution in photon number
In addition to the distribution in energy, it is also useful to know the distribution in
photon number, nκ , for a given mode. This calculation is simpli¬ed by the fact that
the thermal density operator is the product of independent operators for each mode,
ρ= ρκ , (2.173)
κ

where
1
ρκ =
exp (’βNκ ωκ ) . (2.174)

Thus we can drop the mode index and set
ρ = 1 ’ e’β exp ’β ωa† a .
ω
(2.175)
The eigenstates of the single-mode number operator are nondegenerate, so the general
rule (2.132) reduces to
p (n) = Tr (ρ |n n|) = n |ρ| n = 1 ’ e’β e’nβ
ω ω
, (2.176)
where p (n) is the probability of ¬nding n photons. This can be expressed more con-
veniently by ¬rst calculating the average number of photons:
e’β ω

n = Tr ρa a = . (2.177)
1 ’ e’β ω

Using this to eliminate e’β ω
leads to the ¬nal form
n
n
p (n) = . (2.178)
n+1
(1 + n )
Finally, it is important to realize that eqn (2.177) is not restricted to the electro-
magnetic ¬eld. Any physical system with a Hamiltonian of the form (2.89), where the
¼ Quantization of cavity modes

operators a and a† satisfy the canonical commutation relations (2.63) for a harmonic
oscillator, will be described by the Planck distribution.

2.5 Vacuum ¬‚uctuations
Our ¬rst response to the in¬nite zero-point energy associated with vacuum ¬‚uctuations
was to hide it away as quickly as possible, but we now have the tools to investigate
the divergence in more detail. According to eqns (2.99) and (2.100) the electric and
magnetic ¬eld operators are respectively determined by pκ and qκ so there are in-
escapable vacuum ¬‚uctuations of the ¬elds. The E and B ¬elds are linear in aκ and a† κ
so their vacuum expectation values vanish, but E2 and B2 will have nonzero vacuum
expectation values representing the rms deviation of the ¬elds. Let us consider the
rms deviation of the electric ¬eld. The operators Ei (r) (i = 1, 2, 3) are hermitian and
mutually commutative, so we are allowed to consider simultaneous measurements of
all components of E (r). In this case the ambiguity in going from a classical quantity
to the corresponding quantum operator is not an issue.
Since trouble is to be expected, we approach 0 E2 (r) 0 with caution by ¬rst
evaluating 0 |Ei (r) Ej (r )| 0 for r = r. The expansion (2.101) yields

0 |Ei (r) Ej (r )| 0 = ’ ωκ ω» Eκi (r) E»j (r )
2 0 κ »

a» ’ a† |0 ,
— 0| aκ ’ a† (2.179)
κ »

and evaluating the vacuum expectation value leads to

0 |Ei (r) Ej (r )| 0 = ωκ Eκi (r) Eκj (r ) . (2.180)
2 0 κ

Direct evaluation of the sum over modes requires detailed knowledge of both the
mode spectrum and the mode functions, but this can be avoided by borrowing a trick
from quantum mechanics (Cohen-Tannoudji et al., 1977a, Chap. II, Complement B).
According to eqn (2.35) each mode function E κ is an eigenfunction of the operator
’∇2 with eigenvalue kκ . The operator and eigenvalue are respectively mathemati-
2

cal analogues of the kinetic energy operator and the energy eigenvalue in quantum
mechanics (in units such that 2m = = 1). Since ’∇2 is hermitian and E κ is an
eigenfunction, the general argument given in Appendix C.3.6 shows that
1/2
’∇2 Eκ = kκ E κ = kκ E κ .
2 (2.181)

Using this relation, together with ωκ = ckκ , in eqn (2.35) yields
1/2
ωκ Eκi (r) = ckκ Eκi (r) = c kκ Eκi (r) = c ’∇2 Eκi (r) .
2 (2.182)

Thus eqn (2.180) can be replaced by
c 1/2
’∇2 Eκi (r) Eκj (r ) ,
0 |Ei (r) Ej (r )| 0 = (2.183)
2 0 κ
½
Vacuum ¬‚uctuations

which combines with the completeness relation (2.38) to yield

c 1/2
∆⊥ (r ’ r )
0 |Ei (r) Ej (r )| 0 = ’∇2 ij
2 0
d3 k
c ki kj
δij ’ eik·(r’r ) ,
= 3k (2.184)
k2
2 (2π)
0


where the last line follows from the fact that eik·r is an eigenfunction of ’∇2 with
eigenvalue k 2 . Setting r = r and summing over i = j yields the divergent integral

d3 k
0 E2 (r) 0 = k. (2.185)
(2π)3
0

Thus the rms ¬eld deviation is in¬nite at every point r. In the case of the energy this
disaster could be avoided by rede¬ning the zero of energy for each cavity mode, but
no such escape is possible for measurements of the electric ¬eld itself.
This looks a little neater”although no less divergent”if we de¬ne the (volume
averaged) rms deviation by

1
(∆E)2 = d3 rE2 (r) 0 .
0 (2.186)
V V

This is best calculated by returning to eqn (2.180), setting r = r and integrating to
get
2
e2 ,
(∆E) = (2.187)
κ
κ

where the vacuum ¬‚uctuation ¬eld strength, eκ , for mode E κ is

ωκ
eκ = . (2.188)
2 0V
The sum over all modes diverges, but the ¬‚uctuation strength for a single mode is
¬nite and will play an important role in many of the arguments to follow. A similar
calculation for the magnetic ¬eld yields

µ0 ωκ
2
b2 , bκ =
(∆B) = . (2.189)
κ
2V
κ


The source of the divergence in (∆E)2 and (∆B)2 is the singular character of
the the vacuum ¬‚uctuations at a point. This is a mathematical artifact, since any
measuring device necessarily occupies a nonzero volume. This suggests considering an
operator of the form
W ≡’ d3 r P (r) · E (r) , (2.190)
V

where P (r) is a smooth (in¬nitely di¬erentiable) c-number function that vanishes
outside some volume V0 V . In this way, the singular behavior of E (r) is reduced by
¾ Quantization of cavity modes

1/3
averaging the point r over distances of the order d0 = V0 . According to the uncer-
tainty principle, this is equivalent to an upper bound k0 ∼ 1/d0 in the wavenumber,
so the divergent integral in eqn (2.185) is replaced by

d3 k 4
k0
< ∞.
k= (2.191)
(2π)3 2
0 8π
0 k<k0

If the volume V0 is ¬lled with an electret, i.e. a material with permanent electric
polarization, then P (r) can be interpreted as the density of classical dipole moment,
and W is the interaction energy between the classical dipoles and the quantized ¬eld. In
this idealized model W is a well-de¬ned physical quantity which is measurable, at least
in principle. Suppose the measurement is carried out repeatedly in the vacuum state.
According to the standard rules of quantum theory, the average of these measurements
is given by the vacuum expectation value of W , which is zero. Of course, the fact that
the average vanishes does not imply that every measured value does. Let us next
determine the variance of the measurements by evaluating

0| W 2 |0 = d3 r Pi (r) Pj (r ) 0 |Ei (r) Ej (r )| 0 .
d3 r (2.192)
V V

Substituting eqn (2.180) into this expression yields

0| W 2 |0 = ω κ Pκ ,
2
(2.193)
2 0 κ

where
Pκ = d3 r P (r) · E κ (r) (2.194)
V

represents the classical interaction energy for a single mode. In this case the sum
converges, since the coe¬cients Pκ will decay rapidly for higher-order modes. Thus W
exhibits vacuum ¬‚uctuation e¬ects that are both ¬nite and observable. It is important
to realize that this result is independent of the choice of operator ordering, e.g. eqn
(2.105) or eqn (2.106), for the Hamiltonian. It is also important to assume that the
permanent dipole moment of the electret is so small that the radiation it emits by
virtue of the acceleration imparted by the vacuum ¬‚uctuations can be neglected. In
other words, this is a test electret analogous to the test charges assumed in the standard
formulation of classical electrodynamics (Jackson, 1999, Sec. 1.2).

2.6 The Casimir e¬ect
In Section 2.2 we discarded the zero-point energy due to vacuum ¬‚uctuations on the
grounds that it could be eliminated by adding a constant to the Hamiltonian in eqn
(2.104). This is correct for a single cavity, but the situation changes if two di¬erent
cavities are compared. In this case, a single shift in the energy spectrum can eliminate
one or the other, but not both, of the zero-point energies; therefore, the di¬erence
between the zero-point energies of the two cavities can be the basis for observable
¿
The Casimir e¬ect

phenomena. An argument of this kind provides the simplest explanation of the Casimir
e¬ect.
We follow the approach of Milonni and Shih (1992) which begins by considering
the planar cavity”i.e. two plane parallel plates separated by a distance small com-
pared to their lateral dimensions”described in Appendix B.4. In this situation edge
e¬ects are small, so the plates can be represented by an ideal cavity in the shape of a
rectangular box with dimensions L — L — ∆z. This con¬guration will be compared to
a cubical cavity with sides L. The eigenfrequencies for a planar cavity are
1/2
2 2 2
mπ nπ

ωlmn = c + + , (2.195)
L L ∆z

where the range of the indices is l, m, n = 0, 1, 2, . . ., except that there are no modes
with two zero indices. If one index vanishes, there is only one polarization, but for
three nonzero indices there are two. We want to compare the zero-point energies of
con¬gurations with di¬erent values of ∆z, so the interesting quantity is
ωlmn
E0 (∆z) = Clmn , (2.196)
2
l,m,n

where Clmn is the number of polarizations. Thus Clmn = 2 when all indices are nonzero,
Clmn = 1 when exactly one index vanishes, and Clmn = 0 when at least two indices
vanish. Since this sum diverges, it is necessary to regularize it, i.e. to replace it with a
mathematically meaningful expression which has eqn (2.196) as a limiting value. All
intermediate calculations are done using the regularized form, and the limit is taken
at the end of the calculation. The physical justi¬cation for this apparently reckless
procedure rests on the fact that real conductors become transparent to radiation at
su¬ciently high frequencies (Jackson, 1999, Sec. 7.5D). In this range the contribution
to the zero-point energy is unchanged by the presence of the conducting plates, so it
will cancel out in taking the di¬erence between di¬erent con¬gurations. Thus the high-
frequency part of the sum in eqn (2.196) is not physically relevant, and a regularization
scheme that suppresses the contributions of high frequencies can give a physically
meaningful result (Belinfante, 1987).
One regularization scheme is to replace eqn (2.196) by
ωlmn
exp ’±ωlmn Clmn
2
E0 (∆z) = . (2.197)
2
l,m,n

This sum is well behaved for any ± > 0, and approaches the original divergent ex-
pression as ± ’ 0. The energy in a cubical box with sides L is E0 (L) and the ratio
of the volumes is L2 ∆z/L3 = ∆z/L, so the di¬erence between the zero-point energy
contained in the planar box and the zero-point energy contained in the same volume
in the larger box is
∆z
U (∆z) = E0 (∆z) ’ E0 (L) . (2.198)
L
This is just the work done in bringing one of the faces of the cube from the original
distance L to the ¬nal distance ∆z.
Quantization of cavity modes

The regularized sum could be evaluated numerically, but it is more instructive to
exploit the large size of L. In the limit of large L, the sums over l and m in E0 (∆z)
and over all three indices in E0 (L) can be replaced by integrals over k-space according
to the rule (2.169). After a rather lengthy calculation (Milonni and Shih, 1992) one
¬nds
π 2 c L2
U (∆z) = ’ ; (2.199)
720 ∆z 3
consequently, the force attracting the two plates is

π 2 c L2
dU
F =’ =’ . (2.200)
240 ∆z 4
d (∆z)

For numerical estimates it is useful to restate this as
2
0.13 L [cm]
F [µN] = ’ . (2.201)
4
∆z [µm]

For plates with area 1 cm2 separated by 1 µm, the magnitude of the force is 0.13 µN.
This is a very small force; indeed, it is approximately equal to the force exerted by the
proton on the electron in the ¬rst Bohr orbit of a hydrogen atom.
The Casimir force between parallel plates would be extremely hard to measure,
due to the di¬culty of aligning parallel plates separated by 1 µm. Recent experiments
have used a di¬erent con¬guration consisting of a conducting sphere of radius R at a
distance d from a conducting plate (Lamoreaux, 1997; Mohideen and Roy, 1998). For
perfect conductors, a similar calculation yields the force

π3 c R
(d) = ’
(0)
F (2.202)
360 d3
in the limit R d. When corrections for ¬nite conductivity, surface roughness, and
nonzero temperature are included there is good agreement between theory and exper-
iment.
The calculation of the Casimir force sketched above is based on the di¬erence be-
tween the zero-point energies of two cavities, and it provides good agreement between
theory and experiment. This might be interpreted as providing evidence for the real-
ity of zero-point energy, except for two di¬culties. The ¬rst is the general argument
in Section 2.2 showing that it is always permissible to use the normal-ordered form
(2.89) for the Hamiltonian. With this choice, there is no zero-point energy for either
cavity; and our successful explanation evaporates. The second, and more important,
di¬culty is that the forces predicted by eqns (2.200) and (2.202) are independent of
the electronic charge. There is clearly something wrong with this, since all dynamical
e¬ects depend on the interaction of charged particles with the electromagnetic ¬eld.
It has been shown that the second feature is an artifact of the assumption that the
plates are perfect conductors (Ja¬e, 2005). A less idealized calculation yields a Casimir
force that properly vanishes in the limit of zero electronic charge. Thus the agreement
between the theoretical prediction (2.202) and experiment cannot be interpreted as
evidence for the physical reality of zero-point energy. We emphasize that this does
Exercises

not mean that vacuum ¬‚uctuations are not real, since other experiments”such as the
partition noise at beam splitters discussed in Section 8.4.2”do provide evidence for
their e¬ects.
Our freedom to use the normal-ordered form of the Hamiltonian implies that it
must be possible to derive the Casimir force without appealing to the zero point
energy. An approach that does this is based on the van der Waals coupling between
atoms in di¬erent walls. The van der Waals potential can be derived by considering the
coupling between the ¬‚uctuating dipoles of two atoms. This produces a time-averaged
perturbation proportional to (p1 · p2 ) /r3 , where r is the distance between the atoms,
and p1 and p2 are the electric dipole operators. This potential comes from the static
Coulomb interactions between the charged particles comprising the atoms; it does not
involve the radiative modes that contribute to the zero point energy in symmetrical
ordering. The random ¬‚uctuations in the dipole moments p1 and p2 produce no ¬rst-
order correction to the energy, but in second order the dipole“dipole coupling produces
the van der Waals potential VW (r) with its characteristic 1/r7 dependence. The 1/r7
dependence is valid for r »at , where »at is a characteristic wavelength of an atomic
»at the potential varies as 1/r6 .
transition. For r
For many atoms, the simplest assumption is that these potentials are pair-wise
additive, i.e. the total potential energy is

VW (|rn ’ rm |) ,
Vtot = (2.203)
m=n


where the sum runs over all pairs with one atom in each wall. With this approximation,
it is possible to explain about 80% of the Casimir force in eqn (2.200). In fact the
assumption of pair-wise additivity is not justi¬ed, since the presence of a third atom
changes the interaction between the ¬rst two. When this is properly taken into account,
the entire Casimir force is obtained.
Thus there are two di¬erent explanations for the Casimir force, corresponding to
the two choices a† a or a† a + aa† /2 made in de¬ning the electromagnetic Hamil-
tonian. The important point to keep in mind is that the relevant physical prediction”
the Casimir force between the plates”is the same for both explanations. The di¬erence
between the two lies entirely in the language used to describe the situation. This kind
of ambiguity in description is often found in quantum physics. Another example is the
van der Waals potential itself. The explanation given above corresponds to the normal
ordering of the electromagnetic Hamiltonian. If the symmetric ordering is used instead,
the presence of the two atoms induces a change in the zero-point energy of the ¬eld
which becomes increasingly negative as the distance between the atoms decreases. The
result is the same attractive potential between the atoms (Milonni and Shih, 1992).

2.7 Exercises
2.1 Cavity equations
(1) Give the separation of variables argument leading to eqn (2.7).
(2) Derive the equations satis¬ed by E (r) and B (r) and verify eqns (2.9) and (2.10).
Quantization of cavity modes

2.2 Rectangular cavity modes
(1) Use the method of separation of variables to solve eqns (2.11) and (2.1) for a
rectangular cavity, subject to the boundary condition (2.13), and thus verify eqns
(2.14)“(2.17).
(2) Show explicitly that the modes satisfy the orthogonality conditions

d3 rE ks (r) · E k s (r) = 0 for (k, s) = (k , s ) .

(3) Use the normalization condition

d3 rE ks (r) · E ks (r) = 1

to derive the normalization constants Nk .

2.3 Equations of motion for classical radiation oscillators
In the interior of an empty cavity the ¬elds satisfy Maxwell™s equations (2.1) and
(2.2). Use the expansions (2.40) and (2.41) and the properties of the mode functions
to derive eqn (2.43).

2.4 Complex mode amplitudes
(1) Use the expression (2.48) for the classical energy and the expansions (2.40) and
(2.41) to derive eqn (2.49).
(2) Derive eqns (2.46) and (2.51).

2.5 Number states
Use the commutation relations (2.76) and the de¬nition (2.73) of the vacuum state to
verify eqn (2.78).

2.6 The second-order coherence matrix
(1) For the operators a† and ap (p = 1, 2) de¬ned by eqn (2.153) show that the number
p

operators Np = ap ap are simultaneously measurable.
(2) Consider the operator

1 1 1
|1x 1x | + |1y 1y | ’ 1x | + |1x 1y |) ,
(|1y
ρ=
2 2 4

where |1s = a† |0 .
ks

(a) Show that ρ is a genuine density operator, i.e. it is positive and has unit trace.
(b) Calculate the coherence matrix J, its eigenvalues and eigenvectors, and the
degree of polarization.
Exercises

2.7 The Stokes parameters
(1) What is the physical signi¬cance of S0 ?
(2) Use the explicit forms of the Pauli matrices and the expansion (2.158) to show
that
12
S 0 ’ S1 ’ S2 ’ S3 ,
2 2 2
det J =
4
and thereby establish the condition (2.160).
(3) With S0 = 1, introduce polar coordinates by S3 = cos θ, S2 = sin θ sin φ, and
S3 = sin θ cos φ. Find the locations on the Poincar´ sphere corresponding to right
e
circular polarization, left circular polarization, and linear polarization.

2.8 A one-photon mixed state
Consider a monochromatic state for wavevector k (see Section 2.4.1-A) containing
exactly one photon.
(1) Explain why the density operator for this state is completely represented by the
2 — 2 matrix ρss = 1ks |ρ| 1ks .
(2) Show that the density matrix ρss is related to the coherence matrix J by ρss =
Js s .

2.9 The Casimir force
Show that the large L limit of eqn (2.198) is
2
c L
dkx dky e’±k⊥ k⊥
2
U (∆z) =
2 π
2∞
L 1/2
dkx dky e’±(k⊥ +kzn ) k⊥ + kzn
2 2
2 2
+c
π n=1
3
∆z L 2
dkx dky dkz e’±k k ,
’ c
L π
2
2 2 2
where k⊥ = kx + ky , k = k⊥ + kz , and kzn = nπ/∆z.

2.10 Model for the experiments on the Casimir force
Consider the simple-harmonic-oscillator model of the Lamoreaux and Mohideen-Roy
experiments on the Casimir force shown in Fig. 2.1.
All elements of the apparatus, which are assumed to be perfect conductors, are
rendered electrically neutral by grounding them to the Earth. Assume that the spring
constant for the metallic spring is k. (You may ignore Earth™s gravity in this problem.)
(1) Calculate the displacement ∆x of the spring from its relaxed length as a function
of the spacing d between the surface of the sphere and the ¬‚at plate on the right,
after the system has come into mechanical equilibrium.
(2) Calculate the natural oscillation frequency of this system for small disturbances
around this equilibrium as a function of d. Neglect all dissipative losses.
Quantization of cavity modes


Metallic sphere
4
Fig. 2.1 The Casimir force between a
grounded metallic sphere of radius R and the
Metallic spring @
grounded ¬‚at metallic plate on the right, which
Flat metallic plates
is separated by a distance d from the sphere,
can be measured by measuring the displace-
Earth ground
ment of the metallic spring. (Ignore gravity.)


(3) Plot your answers for parts 1 and 2 for the following numerical parameters:

R = 200 µm ,
0.1 µm d 1.0 µm ,
k = 0.02 N/m .
3
Field quantization

Quantizing the radiation oscillators associated with the classical modes of the elec-
tromagnetic ¬eld in a cavity provides a satisfactory theory of the Planck distribution
and the Casimir e¬ect, but this is only the beginning of the story. There are, after all,
quite a few experiments that involve photons propagating freely through space, not
just bouncing back and forth between cavity walls. In addition to this objection, there
is a serious ¬‚aw in the cavity-based model. The quantized radiation oscillators are
de¬ned in terms of a set of classical mode functions satisfying the idealized boundary
conditions for perfectly conducting walls. This di¬culty cannot be overcome by sim-
ply allowing for ¬nite conductivity, since conductivity is itself a macroscopic property
that does not account for the atomistic structure of physical walls. Thus the quantiza-
tion conjecture (2.61) builds the idealized macroscopic boundary conditions into the
foundations of the microscopic quantum theory of light. A fundamental microscopic
theory should not depend on macroscopic idealizations, so there is more work to be
done. We should emphasize, however, that this objection to the cavity model does not
disqualify it as a guide toward an improved theory. The cavity model itself was con-
structed by applying the ideas of nonrelativistic quantum mechanics to the classical
radiation oscillators. In a similar fashion, we will use the cavity model to suggest a
true microscopic conjecture for the quantization of the electromagnetic ¬eld.
In the following sections we will show how the quantization scheme of the cavity
model can be used to suggest local commutation relations for quantized ¬elds in free
space. The experimentally essential description of photons in passive optical devices
will be addressed by formulating a simple model for the quantization of the ¬eld in
a dielectric medium. In the ¬nal four sections we will discuss some more advanced
topics: the angular momentum of light, a description of quantum ¬eld theory in terms
of wave packets, and the question of the spatial localizability of photons.



3.1 Field quantization in the vacuum
The quantization of the electromagnetic ¬eld in free space is most commonly carried
out in the language of canonical quantization (Cohen-Tannoudji et al., 1989, Sec.
II.A.2), which is based on the Lagrangian formulation of classical electrodynamics.
This is a very elegant way of packaging the necessary physical conjectures, but it
requires extra mathematical machinery that is not needed for most applications. We
will pursue a more pedestrian route which builds on the quantization rules for the
ideal physical cavity. To this end, we initially return to the cavity problem.
¼ Field quantization

3.1.1 Local commutation relations
In Chapter 2 we concentrated on the operators (qκ , pκ ) for a single mode. Since the
modes are determined by the boundary conditions at the cavity walls, they describe
global properties of the cavity. We now want turn attention away from the overall
properties of the cavity, in order to concentrate on the local properties of the ¬eld
operators. We will do this by combining the expansions (2.99) and (2.103) for the time-
independent, Schr¨dinger-picture operators E (r) and A (r) with the commutation
o
relations (2.61) for the mode operators to calculate the commutators between ¬eld
components evaluated at di¬erent points in space. The expansions show that E (r)
only depends on the pκ s while A (r) and B (r) depend only on the qκ s; therefore, the
commutation relations, [pκ , p» ] = [qκ , q» ] = 0, produce

[Ej (r) , Ek (r )] = 0 , [Aj (r) , Ak (r )] = 0 , [Bj (r) , Bk (r )] = 0 . (3.1)

On the other hand, [qκ , p» ] = i δκ» , so the commutator between the electric ¬eld and
the vector potential is
1
[Ai (r) , ’Ej (r )] = [qκ , p» ] Eκi (r) E»j (r )
0 κ »
i
Eκi (r) Eκj (r ) .
= (3.2)
0 κ

For any cavity, the mode functions satisfy the completeness condition (2.38), so we see
that
i
[Ai (r) , ’Ej (r )] = ∆⊥ (r ’ r ) . (3.3)
ij
0

The resemblance between this result and the canonical commutation relation, [qκ , p» ] =
i δκ» , for the mode operators suggests the identi¬cation of A (r) and ’E (r) as the
canonical variables for the ¬eld in position space. A similar calculation for the commu-
tator between the E- and B-¬elds can be carried out using eqn (2.100), or by applying
the curl operation to eqn (3.3), with the result

ijl ∇l δ (r ’ r ),
[Bi (r) , Ej (r )] = i (3.4)
0

where ijl is the alternating tensor de¬ned by eqn (A.3). The uncertainty relations
implied by the nonvanishing commutators between electric and magnetic ¬eld compo-
nents were extensively studied in the classic work of Bohr and Rosenfeld (1950), and
a simple example can be found in Exercise 3.2.
The derivation of the local commutation relations (3.1) and (3.3) for the ¬eld oper-
ators in the physical cavity employs the complete set of cavity modes, which depend on
the geometry of the cavity. This can be seen from the explicit appearance of the mode
functions in the second line of eqn (3.2). However, the ¬nal result (3.3) follows from the
completeness relation (2.38), which has the same form for every cavity. This feature
only depends on the fact that the boundary conditions guarantee the Hermiticity of
the operator ’∇2 . We have, therefore, established the quite remarkable result that
½
Field quantization in the vacuum

the local position-space commutation relations are independent of the shape and size
of the cavity. In particular, eqns (3.1) and (3.3) will hold in the limit of an in¬nitely
large physical cavity; that is, when the distance to the cavity walls from either of the
points r and r is much greater than any physically relevant length scale. In this limit,
it is plausible to assume that the boundary conditions at the walls are irrelevant. This
suggests abandoning the original quantization conjecture (2.61), and replacing it by
eqns (3.1) and (3.3). In this way we obtain a microscopic theory which does not involve
the macroscopic idealizations associated with the classical boundary conditions. We
emphasize that this is not a derivation of the local commutation relations from the
physical cavity relations (2.61). The sole function of the cavity-based calculation is
to suggest the form of eqns (3.1) and (3.3), which constitute an independent quanti-
zation conjecture. As always, the validity of the this conjecture has to be tested by
means of experiment. In this new approach, the theory based on the ideal physical
cavity”with its dependence on macroscopic boundary conditions”is demoted to a
phenomenological model.
Since the new quantization rules hold everywhere in space, they can be expressed
in terms of Fourier transform pairs de¬ned by
d3 k
d3 re’ik·r F (r) ,
ik·r
F (r) = 3e F (k) , F (k) = (3.5)
(2π)
where F = A, E, or B. The position-space ¬eld operators are hermitian, so their
Fourier transforms satisfy F† (k) = F (’k). It should be clearly understood that eqn
(3.5) is simply an application of the Fourier transform; no additional physical assump-
tions are required. By contrast, the expansions (2.99) and (2.103) in cavity modes
involve the idealized boundary conditions at the cavity walls.
Transforming eqns (3.1) and (3.3) with respect to r and r independently yields
the equivalent relations
[Ej (k) , Ek (k )] = [Aj (k) , Ak (k )] = 0 , (3.6)
and
i
∆⊥ (k) (2π) δ k + k ,
3
[Ai (k) , ’Ej (k )] = (3.7)
ij
0
where the delta function comes from using the identity (A.96).

3.1.2 Creation and annihilation operators
A Position space
The commutation relations (3.1)“(3.4) are not the only general consequences that are
implied by the cavity model. For example, the expansions (2.101) and (2.103) can be
rewritten as
E (r) = E(+) (r) + E(’) (r) , A (r) = A(+) (r) + A(’) (r) , (3.8)
where
aκ E κ (r) = A(’)† (r)
A(+) (r) = (3.9)
2 0 ωκ
κ
¾ Field quantization

and
ωκ
aκ E κ (r) = E(’)† (r) .
E(+) (r) = i (3.10)
20
κ

Let F be one of the ¬eld operators, Ai or Ei , then F (+) is called the positive-frequency
part and F (’) is called the negative-frequency part. The origin of these mysterious
names will become clear in Section 3.2.3, but for the moment we only need to keep
in mind that F (+) is a sum of annihilation operators and F (’) is a sum of creation
operators. These properties are expressed by

F (+) (r) |0 = 0 , 0| F (’) (r) = 0 . (3.11)

In view of the de¬nition (3.9) there is a natural inclination to think of A(+) (r) as an
operator that annihilates a photon at the point r, but this temptation must be resisted.
The di¬culty is that the photon”i.e. ˜a quantum of excitation of the electromagnetic
¬eld™”cannot be sharply localized in space. A precise interpretation for A(+) (r) is
presented in Section 3.5.2, and the question of photon localization is studied in Section
3.6.
An immediate consequence of eqns (3.9) and (3.10) is that

F (±) (r) , G(±) (r ) = 0 , (3.12)

where F and G are any pair of ¬eld operators. It is clear, however, that F (+) , G(’)
will not always vanish. In particular, a calculation similar to the one leading to eqn
(3.3) yields
i
∆⊥ (r ’ r ) .
(+) (’)
Ai (r) , ’Ej (r ) = (3.13)
2 0 ij
The decomposition (3.8) also allows us to express all ¬eld operators in terms of
(±)
A . For this purpose, we rewrite eqn (3.10) as

aκ kκ E κ (r) ,
E(+) (r) = ic (3.14)
2 0 ωκ
κ

and use eqn (2.181) to get the ¬nal form
1/2
E(+) (r) = ic ’∇2 A(+) (r) . (3.15)

Substituting this into eqn (3.13) yields the equivalent commutation relations

’1/2
∆⊥ (r ’ r ) ,
(+) (’)
’∇2
Ai (r) , Aj (r ) = (3.16)
ij
2 0c
c 1/2
∆⊥ (r ’ r ) .
(+) (’)
’∇2
Ei (r) , Ej (r ) = (3.17)
ij
20
’1/2
1/2
In the context of free space, the unfamiliar operators ’∇2 and ’∇2 are
best de¬ned by means of Fourier transforms. For any real function f (u) the identity
¿
Field quantization in the vacuum

’∇2 exp (ik · r) = k 2 exp (ik · r) allows us to de¬ne the action of f ’∇2 on a plane
wave by f ’∇2 eik·r ≡ f k 2 eik·r . This result in turn implies that f ’∇2 acts on
a general function • (r) according to the rule

d3 k d3 k
f ’∇2 • (r) ≡ 3 • (k) f ’∇
2
eik·r = k 2 eik·r .
3 • (k) f (3.18)
(2π) (2π)

After using the inverse Fourier transform on • (k) this becomes

f ’∇2 • (r) = d3 r r f ’∇2 r • (r ) , (3.19)

where
d3 k
k 2 eik·(r’r )
r f ’∇2 r = 3f (3.20)
(2π)

is the integral kernel de¬ning f ’∇2 as an operator in r-space. Despite its abstract
appearance, this de¬nition is really just a labor saving device; it avoids transforming
back and forth from position space to reciprocal space. For example, real functions of
the hermitian operator ’∇2 are also hermitian; so one gets a useful integration-by-parts
identity

d3 rψ — (r) f ’∇2 • (r) = d3 r f ’∇2 ψ — (r) • (r) , (3.21)

without any intermediate steps involving Fourier transforms.
The equations (3.8), (3.11)“(3.13), (3.15), and (3.16) were all derived by using the
expansions of the ¬eld operators in cavity modes, but once again the ¬nal forms are
independent of the size and shape of the cavity. Consequently, these results are valid
in free space.

B Reciprocal space
The rather strange looking result (3.16) becomes more understandable if we note that
the decomposition (3.8) into positive- and negative-frequency parts applies equally
well in reciprocal space, so that A (k) = A(+) (k) + A(’) (k). The Fourier transforms
of eqns (3.12) and (3.16) with respect to r and r yield respectively

(±) (±)
Ai (k) , Aj (k ) = 0 (3.22)

and
∆⊥ (k)
(+) (’) ij
(2π)3 δ k ’ k .
Ai (k) , Aj (’k ) = (3.23)
2 0c k
This reciprocal-space commutation relation does not involve any strange operators, like
’1/2
’∇2 , but it is still rather complicated. Simpli¬cation can be achieved by noting
Field quantization

that the circular polarization unit vectors eks ”see Appendix B.3.2”are eigenvectors
of ∆⊥ (k) with eigenvalue unity:
ij


∆⊥ (k) (eks )j = (eks )i . (3.24)
ij

By forming the inner product of both sides of eqns (3.22) and (3.23) with e— and
ks
ek s and remembering F† (k) = F (’k), one ¬nds

[as (k) , as (k )] = a† (k) , a† (k ) = 0 (3.25)
s s


and
as (k) , a† (k ) = δss (2π) δ k ’ k ,
3
(3.26)
s

where
2 0 ωk
e— · A(+) (k)
as (k) = (3.27)
ks

and ωk = ck. The operators as (k), combined with the Fourier transform relation (3.5),
provide a replacement for the cavity-mode expansions (3.9) and (3.10):

d3 k
(+)
as (k) eks eik·r ,
A (r) = (3.28)
3 2 0 ωk
(2π) s

d3 k ωk
E(+) (r) = i as (k) eks eik·r . (3.29)
3 20
(2π) s

The number operator

d3 k
a† (k) as (k)
N= (3.30)
s
3
(2π) s

satis¬es
N, a† (k) = a† (k) , [N, as (k)] = ’as (k) , (3.31)
s s

and the vacuum state is de¬ned by as (k) |0 = 0, so it seems that a† (k) and as (k) can
s
be regarded as creation and annihilation operators that replace the cavity operators
aκ and a† . However, the singular commutation relation (3.26) exacts a price. For
κ
example, the one-photon state |1ks = a† (k) |0 is an improper state vector satisfying
s
the continuum normalization conditions
3
1k s |1ks = δss (2π) δ k ’ k . (3.32)

Thus a properly normalized one-photon state is a wave packet state

d3 k
¦s (k) a† (k) |0 ,
|¦ = (3.33)
s
3
(2π) s

where the c-number function ¦s (k) is normalized by
Field quantization in the vacuum

d3 k 2
|¦s (k)| = 1 . (3.34)
3
(2π) s

The Fock space HF consists of all linear combinations of number states,

d3 k1 d3 kn
¦s1 ···sn (k1 , . . . , kn ) a†1 (k1 ) · · · a†n (kn ) |0 ,
|¦ = ··· ··· s s
3 3
(2π) (2π) s1 sn
(3.35)
where
d3 k1 d3 kn
|¦s1 ···sn (k1 , . . . , kn )|2 < ∞
··· ··· (3.36)
3 3
(2π) (2π) s1 sn

and n = 0, 1, . . ..

3.1.3 Energy, momentum, and angular momentum
A The Hamiltonian
The expression (2.105) for the ¬eld energy in a cavity can be converted to a form
suitable for generalization to free space by ¬rst inverting eqn (3.10) to get

20
aκ = ’i d3 rE κ (r) · E(+) (r) . (3.37)
ωκ V

The next step is to substitute this expression for aκ into eqn (2.105) and carry out the
sum over κ by means of the completeness relation (2.38); this calculation leads to

ω κ a† aκ
Hem = κ
κ

(r) ∆⊥ (r ’ r ) Ej
(’) (+)
d3 r d3 r Ei
=2 (r ) . (3.38)
0 ij
V V

Since the free-¬eld operator E(+) (r ) is transverse, the in¬nite volume limit is

d3 rE(’) (r) · E(+) (r) .
Hem = 2 (3.39)
0


This can also be expressed as

d3 rA(’) (r) · ’∇2 A(+) (r) ,
Hem = 2 0 c2 (3.40)

by using eqn (3.15). A more intuitively appealing form is obtained by using the plane-
wave expansion (3.29) for E(±) to get

d3 k
a† (k) as (k) .
Hem = ωk (3.41)
s
3
(2π) s
Field quantization

B The linear momentum
The cavity model does not provide any expressions for the linear momentum and
the angular momentum, so we need independent arguments for them. The reason for
the absence of these operators is the presence of the cavity walls. From a mechanical
point of view, the linear momentum and the angular momentum of the ¬eld are not
conserved because of the immovable cavity. Alternatively, we note that one of the
fundamental features of quantum theory is the identi¬cation of the linear momentum
and the angular momentum operators with the generators for spatial translations and
rotations respectively (Bransden and Joachain, 1989, Secs 5.9 and 6.2). This means
that the mechanical conservation laws for linear and angular momentum are equivalent
to invariance under spatial translations and rotations respectively. The location and
orientation of the cavity in space spoils both invariances.
Since the cavity model fails to provide any guidance, we once again call on the
correspondence principle by quoting the classical expression for the linear momentum
(Jackson, 1999, Sec. 6.7):

P= 0E ⊥ —B
d3 r

0E — (∇ — A) .
d3 r
= (3.42)

The vector identity F— (∇ — G) = Fj ∇Gj ’ (F · ∇) G combined with an integration
by parts and the transverse nature of E (r) provides the more useful expression

P= d3 rEj (r) ∇Aj (r) . (3.43)
0


The initial step in constructing the corresponding Schr¨dinger-picture operator is to
o
replace the classical ¬elds according to

A (r) ’ A (r) = A(+) (r) + A(’) (r) , (3.44)
E (r) ’ E (r) = E(+) (r) + E(’) (r) . (3.45)

The momentum operator P is then the sum of four terms, P = P(+,+) + P(’,’) +
P(’,+) + P(+,’) , where

(σ) („ )
d3 rEj ∇Aj for σ, „ = ± .
P(σ,„ ) = (3.46)
0


Each of these terms is evaluated by using the plane-wave expansions (3.28) and (3.29),
together with the orthogonality relation, e— ·eks = δss , and the re¬‚ection property”
ks
see eqn (B.73)”e’k,s = e— , (s = ±) for the circular polarization basis. The ¬rst result
ks
is P(+,+) = P(’,’)† = 0; consequently, only the cross terms survive to give

d3 k k
a† (k) as (k) + as (k) a† (k) .
P= (3.47)
s s
3 2
(2π) s
Field quantization in the vacuum

This is analogous to the symmetrical ordering (2.106) for the Hamiltonian in the cavity
problem, so our previous experience suggests replacing the symmetrical ordering by
normal ordering, i.e.
d3 k
a† (k) as (k) .
P= k (3.48)
s
3
(2π) s

From this expression and eqn (3.41), it is easy to see that [P, Hem ] = 0 and [Pi , Pj ] = 0.
Any observable commuting with the Hamiltonian is called a constant of the motion,
so the total momentum is a constant of the motion and the individual components Pi
are simultaneously measurable.
By using the inverse Fourier transform,

2 0 ωk
e— · d3 re’ik·r A(+) (r) ,
as (k) = (3.49)
ks


which is the free-space replacement for eqn (3.37), or proceeding directly from eqn
(3.46), one ¬nds the equivalent position-space representation

(’) (+)
(r) ∇Aj
d3 rEj
P=2 (r) . (3.50)
0



C The angular momentum
Finally we turn to the classical expression for the angular momentum (Jackson, 1999,
Sec. 12.10):

J= d3 r r — [ 0 E (r, t) — B (r, t)] . (3.51)

Combining B = ∇ — A with the identity F — (∇ — G) = Fj ∇Gj ’ (F · ∇) G allows
this to be written in the form J = L + S, where

L= d3 r Ej (r — ∇) Aj (3.52)
0



and
S= d3 r E — A . (3.53)
0


Once again, the initial guess for the corresponding quantum operators is given by
applying the rules (3.44) and (3.45), so the total angular momentum operator is

J = L + S, (3.54)

where the operators L and S are de¬ned by quantizing the classical expressions L and
S respectively.
Field quantization

The application of the method used for the linear momentum to eqn (3.52) is com-
plicated by the explicit r-term, but after some e¬ort one ¬nds the rather cumbersome
expression (Simmons and Guttmann, 1970)

d3 k
i ‚
Mi† (k) k —
L=’ Mi (k) ’ HC
3
2 ‚k
(2π)
d3 k ‚

= ’i (k) k —
3 Mi Mi (k) , (3.55)
‚k
(2π)
where
M (k) = as (k) eks . (3.56)
s

In this case a substantial simpli¬cation results from translating the reciprocal-space
representation back into position space to get
2i 0 (’) (+)
(r) r — ∇ Aj
d3 rEj
L= (r) . (3.57)
i
A straightforward calculation using eqn (3.39) shows that L is also a constant of the
motion, i.e. [L, Hem ] = 0. However, the components of L are not mutually commuta-
tive, so they cannot be measured simultaneously.
The quantization of eqn (3.53) goes much more smoothly, and leads to the normal-
ordered expression
d3 k
sa† (k) as (k)
S= 3k s
(2π) s
3
dk
a† (k) a+ (k) ’ a† (k) a’ (k) ,
= 3k (3.58)

+
(2π)

where k = k/k is the unit vector along k. Another use of eqn (3.49) yields the equiv-
alent position-space form

d3 rE(’) — A(+) .
S=2 (3.59)
0


The expression (3.54) for the total angular momentum operator looks like the
decomposition into orbital and spin parts familiar from quantum mechanics, but this
resemblance is misleading. For the electromagnetic ¬eld, the interpretation of eqn
(3.54) poses a subtle problem which we will take up in Section 3.4.

D The helicity operator
It is easy to show that S commutes with P and with Hem , and further that

[Si , Sj ] = 0 . (3.60)

Thus S, P, and Hem are simultaneously measurable, and there are simultaneous eigen-
vectors for them. In the simplest case of the improper one-photon state |1ks =
Field quantization in the vacuum


a† (k) |0 , one ¬nds: Hem |1ks = ωk |1ks , P |1ks = k |1ks , k — S |1ks = 0, and
s
k · S |1ks = s |1ks . Thus |1ks is an eigenvector of the longitudinal component k · S
with eigenvalue s and an eigenvector of the transverse components k — S with eigen-
value 0. For the circular polarization basis, the index s represents the helicity, so S is
called the helicity operator.

E Evidence for helicity and orbital angular momentum
Despite the conceptual di¬culties mentioned in Section 3.1.3-C, it is possible to devise
experiments in which certain components of the helicity S and the orbital angular mo-
mentum L are separately observed. The ¬rst measurement of this kind (Beth, 1936)
was carried out using an experimental arrangement consisting of a horizontal wave
plate suspended at its center by a torsion ¬ber, so that the plate is free to undergo
twisting motions around the vertical axis. In a simpli¬ed version of this experiment,
a vertically-directed, linearly-polarized beam of light is allowed to pass through a
quarter-wave plate, which transforms it into a circularly-polarized beam of light (Born
and Wolf, 1980, Sec. 14.4.2). Since the experimental setup is symmetrical under ro-
tations around the vertical axis (the z-axis), the z-component of the total angular
momentum will be conserved.
We will use a one-photon state

ξs a† (k) |0 ,
|ψ = ξs |1ks = (3.61)
s
s s

with k = ku3 directed along the z-axis, as a simple model of an incident light beam
of arbitrary polarization. A straightforward calculation using eqn (3.55) for Lz shows
that Lz |1ks = 0; consequently, Lz |ψ = 0 for any choice of the coe¬cients ξs . In
other words, states of this kind have no z-component of orbital angular momentum.
The particular choice
1 1
= √ [|1k+ + |1k’ ] = √ a† (k) + a† (k) |0
|ψ (3.62)

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