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)

Zκ

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

lπ

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