Fi (r) = Fj (r ) .

d3 r (Eks (r))i (Eks (r ))j (2.31)

V ks

On the other hand, eqn (2.24) leads to

Gj (r ) = 0

d3 r (Eks (r))i (Eks (r ))j (2.32)

V ks

for any longitudinal ¬eld G (r). Thus the integral operator de¬ned by the expression

in curly brackets annihilates longitudinal ¬elds and reproduces transverse ¬elds. Two

operators that have the same action on the entire space of vector ¬elds are identical;

therefore,

(Eks (r))i (Eks (r ))j = ∆⊥ (r ’ r ) . (2.33)

ij

ks

A similar argument applied to the magnetic mode functions leads to the corresponding

result:

(Bks (r))i (Bks (r ))j = ∆⊥ (r ’ r ) . (2.34)

ij

ks

¿

Quantization of cavity modes

C The general cavity

Now that we have mastered the simple rectangular cavity, we proceed to a general

metallic cavity with a bounding surface S of arbitrary shape.3 As we have already

remarked, the di¬erence between this general cavity and the rectangular cavity lies

entirely in the boundary conditions. The solution of the Helmholtz equations (2.11)

and (2.12), together with the general boundary conditions (2.5) and (2.6), has been

extensively studied in connection with the theory of microwave cavities (Slater, 1950).

Separation of variables is not possible for general boundary shapes, so there is no

way to obtain the explicit solutions shown in Section 2.1.1-A. Fortunately, we only

need certain properties of the solutions, which can be obtained without knowing the

explicit forms. General results from the theory of partial di¬erential equations (Za-

uderer, 1983, Sec. 8.1) guarantee that the Helmholtz equation in any ¬nite cavity

has a complete, orthonormal set of eigenfunctions labeled by a discrete multi-index

κ = (κ1 , κ2 , κ3 , κ4 ) that replaces the combination (k, s) used for the rectangular cav-

ity. These normal mode functions E κ (r) and Bκ (r) are real, transverse vector

¬elds satisfying the boundary conditions (2.5) and (2.6) respectively, together with

the Helmholtz equation:

∇2 + kκ E κ = 0 ,

2

(2.35)

∇2 + kκ Bκ = 0 ,

2

(2.36)

where kκ = ωκ /c and ωκ is the cavity resonance frequency of mode κ. The allowed val-

ues of the discrete indices κ1 , . . . , κ4 and the resonance frequencies ωκ are determined

by the geometrical properties of the cavity.

By combining the orthonormality conditions

d3 rE κ · E » = δκ» ,

V

(2.37)

d rBκ · B» = δκ»

3

V

with the completeness of the modes, we can repeat the argument in Section 2.1.1-B

to obtain the general completeness identities

Eκi (r) Eκj (r ) = ∆⊥ (r ’ r ) , (2.38)

ij

κ

Bκi (r) Bκj (r ) = ∆⊥ (r ’ r ) . (2.39)

ij

κ

D The classical electromagnetic energy

Since the cavity mode functions are a complete orthonormal set, general electric and

magnetic ¬elds”and the associated vector potential”can be written as

3 Theterm ˜arbitrary™ should be understood to exclude topologically foolish choices, such as re-

placing the rectangular cavity by a Klein bottle.

¿ Quantization of cavity modes

1

E (r, t) = ’ √ Pκ (t) E κ (r) , (2.40)

0 κ

√

B (r, t) = µ0 ωκ Qκ (t) Bκ (r) , (2.41)

κ

1

A (r, t) = √ Qκ (t) E κ (r) . (2.42)

0 κ

Substituting the expansions (2.40) and (2.41) into the vacuum Maxwell equations

(2.1)“(2.4) leads to the in¬nite set of ordinary di¬erential equations

™ ™

Qκ = Pκ and Pκ = ’ωκ Qκ .

2

(2.43)

For each mode, this pair of equations is mathematically identical to the equations of

motion of a simple harmonic oscillator, where the expansion coe¬cients Qκ and Pκ

respectively play the roles of the oscillator coordinate and momentum. On the basis

of this mechanical analogy, the mode κ is called a radiation oscillator, and the set

of points

{(Qκ , Pκ ) for ’∞ < Qκ < ∞ and ’∞ < Pκ < ∞} (2.44)

is said to be the classical oscillator phase space for the κth mode.

For the transition to quantum theory, it is useful to introduce the dimensionless

complex amplitudes

ωκ Qκ (t) + iPκ (t)

√

±κ (t) = , (2.45)

2 ωκ

which allow the pair of real equations (2.43) to be rewritten as a single complex

equation,

±κ (t) = ’iωκ ±κ (t) ,

™ (2.46)

with the general solution ±κ (t) = ±κ e’iωκ t , ±κ = ±κ (0). The expansions for the ¬elds

can all be written in terms of ±κ and ±— ; for example eqn (2.40) becomes

κ

ωκ

±κ e’iωκ t E κ (r) + CC .

E (r, t) = i (2.47)

20

κ

One of the chief virtues of the expansions (2.40) and (2.41) is that the orthogonality

relations (2.37) allow the classical electromagnetic energy in the cavity,

1

+ µ’1 B2 ,

Uem = 0E

2

d3 r (2.48)

0

2 V

to be expressed as a sum of independent terms: one for each normal mode,

1

Uem = Pκ + ωκ Q2 .

2 2

(2.49)

κ

2

κ

Each term in the sum is mathematically identical to the energy of a simple harmonic

oscillator with unit mass, oscillator frequency ωκ , coordinate Qκ , and momentum Pκ .

For each κ, eqn (2.43) is obtained from

¿

Quantization of cavity modes

‚Uem ‚Uem

™ ™

and Pκ = ’

Qκ = ; (2.50)

‚Pκ ‚Qκ

consequently, Uem serves as the classical Hamiltonian for the radiation oscillators,

and Qκ and Pκ are said to be canonically conjugate classical variables (Marion

and Thornton, 1995). An even more suggestive form comes from using the complex

amplitudes ±κ to write the energy as

ωκ ±— ±κ .

Uem = (2.51)

κ

κ

Interpreting ±— ±κ as the number of light-quanta with energy ωκ makes this a real-

κ

ization of Einstein™s original model.

2.1.2 The quantization conjecture

The simple harmonic oscillator is one of the very few examples of a mechanical sys-

tem for which the Schr¨dinger equation can be solved exactly. For a classical me-

o

chanical oscillator, Q (t) represents the instantaneous displacement of the oscillating

mass from its equilibrium position, and P (t) represents its instantaneous momentum.

The trajectory {(Q (t) , P (t)) for t 0} is uniquely determined by the initial values

(Q, P ) = (Q (0) , P (0)).

The quantum theory of the mechanical oscillator is usually presented in the coor-

dinate representation, i.e. the state of the oscillator is described by a wave function

ψ (Q, t), where the argument Q ranges over the values allowed for the classical co-

ordinate. Thus the wave functions belong to the Hilbert space of square-integrable

2

functions on the interval (’∞, ∞). In the Born interpretation, |ψ (Q, t)| represents

the probability density for ¬nding the oscillator with a displacement Q from equilib-

rium at time t; consequently, the wave function satis¬es the normalization condition

∞

2

dQ |ψ (Q, t)| = 1 . (2.52)

’∞

In this representation the classical oscillator variables (Q, P )”representing the pos-

sible initial values of classical trajectories”are replaced by the quantum operators q

and p de¬ned by

‚

qψ (Q, t) = Qψ (Q, t) and pψ (Q, t) = ψ (Q, t) . (2.53)

i ‚Q

By using the explicit de¬nitions of q and p it is easy to show that the operators satisfy

the canonical commutation relation

[q, p] = i . (2.54)

For a system consisting of N noninteracting mechanical oscillators”with coordi-

nates Q1 , Q2 , . . . , QN ”the coordinate representation is de¬ned by the N -body wave

function

ψ (Q1 , Q2 , . . . , QN , t) , (2.55)

¼ Quantization of cavity modes

and the action of the operators is

qm ψ (Q1 , Q2 , . . . , QN , t) = Qm ψ (Q1 , Q2 , . . . , QN , t) ,

(2.56)

‚

pm ψ (Q1 , Q2 , . . . , QN , t) = ψ (Q1 , Q2 , . . . , QN , t) ,

i ‚Qm

where m = 1, . . . , N . This explicit de¬nition, together with the fact that the Qm s

are independent variables, leads to the general form of the canonical commutation

relations,

[qm , pm ] = i δmm , (2.57)

[qm , qm ] = [pm , pm ] = 0 , (2.58)

for m, m = 1, . . . , N . This mechanical system is said to have N degrees of freedom.

The results of the previous section show that the pairs of coe¬cients (Qκ , Pκ )

in the expansions (2.40) and (2.41) are canonically conjugate and that they satisfy

the same equations of motion as a mechanical harmonic oscillator. Since the classical

descriptions of the radiation and mechanical oscillators have the same mathematical

form, it seems reasonable to conjecture that their quantum theories will also have the

same form. For the κth cavity mode this simply means that the state of the radiation

oscillator is described by a wave function ψ (Qκ , t). In order to distinguish between

the radiation and mechanical oscillators, we will call the quantum operators for the

radiation oscillator qκ and pκ . The mathematical de¬nitions of these operators are still

given by eqn (2.56), with qκ and pκ replaced by qκ and pκ .

Extending this procedure to describe the general state of the cavity ¬eld introduces

a new complication. The classical state of the electromagnetic ¬eld is represented by

functions E (r, t) and B (r, t) that, in general, cannot be described by a ¬nite number

of modes. This means that the classical description of the cavity ¬eld requires in¬-

nitely many degrees of freedom. A naive interpretation of the quantization conjecture

would therefore lead to wave functions ψ (Q1 , Q2 , . . .) that depend on in¬nitely many

variables. Mathematical techniques to deal with such awkward objects do exist, but it

is much better to start with abstract algebraic operator relations like eqns (2.57) and

(2.58), and then to choose an explicit representation that is well suited to the problem

at hand.

The formulation of quantum mechanics used above is called the Schr¨dinger o

picture; it is characterized by time-dependent wave functions and time independent

operators. The Schr¨dinger-picture formulation of the quantization conjecture for the

o

electromagnetic ¬eld therefore consists of the following two parts.

(1) The time-dependent states of the electromagnetic ¬eld satisfy the superposition

principle: if |Ψ (t) and |¦ (t) are two physically possible states, then the super-

position

± |Ψ (t) + β |¦ (t) (2.59)

is also a physically possible state. (See Appendix C.1 for the bra and ket notation.)

½

Quantization of cavity modes

(2) The classical variables Qκ = Qκ (t = 0) and Pκ = Pκ (t = 0) are replaced by time-

independent hermitian operators qκ and pκ :

Qκ ’ qκ and Pκ ’ pκ , (2.60)

that satisfy the canonical commutation relations

[qκ , pκ ] = i δκκ , [qκ , qκ ] = 0 , and [pκ , pκ ] = 0 , (2.61)

where κ, κ range over all cavity modes.

The statements (1) and (2) are equally important parts of this conjecture.

Another useful form of the commutation relations (2.61) is provided by de¬ning

the dimensionless, non-hermitian operators

ωκ qκ ’ ipκ

ωκ qκ + ipκ

and a† = √

√

aκ = (2.62)

κ

2 ωκ 2 ωκ

for the κth mode of the radiation ¬eld. A simple calculation using eqn (2.61) yields

the equivalent commutation relations

aκ , a† = δκκ , [aκ , aκ ] = 0 . (2.63)

κ

To sum up: by examining the problem of the ideal resonant cavity, we have been

led to the conjecture that the radiation ¬eld can be viewed as a collection of quantized

simple harmonic oscillators. The quantization conjecture embodied in eqns (2.59)“

(2.61) may appear to be rather formal and abstract, but it is actually the fundamental

physical assumption required for constructing the quantum theory of the electromag-

netic ¬eld. New principles of this kind cannot be deduced from the pre-existing theory;

instead, they represent a genuine leap of scienti¬c induction that must be judged by

its success in explaining experimental results.

In the following section, we will combine the canonical commutation relations with

some basic physical principles to construct the Hilbert space of state vectors |Ψ , and

thus obtain a concrete representation of the operators qκ and pκ or aκ and a† for aκ

single cavity mode. In Section 2.1.2-C this representation will be generalized to include

the in¬nite set of normal cavity modes.

A The single-mode Fock space

In this section we will deal with a single mode, so the mode index can be omitted.

Instead of starting with the coordinate representation of the wave function, as in eqn

(2.53), we will deduce the structure of the Hilbert space of states by the following

argument. According to eqn (2.49) the classical energy for a single mode is

1

Uem = P 2 + ω 2 Q2 , (2.64)

2

where the arbitrary zero of energy has been chosen to correspond to the classical

solution Q = P = 0, representing the oscillator at rest at the minimum of the potential.

¾ Quantization of cavity modes

In quantum mechanics the standard procedure is to apply eqn (2.60) to this expression

and to interpret the resulting operator as the (single-mode) Hamiltonian

12

p + ω2 q2 .

Hem = (2.65)

2

It is instructive to rewrite this in terms of the operators a and a† by solving eqn (2.62)

to get

ω

a + a† and p = ’i a ’ a† .

q= (2.66)

2ω 2

Substituting these expressions into eqn (2.65)”while remembering that the operators

a and a† do not commute”leads to

1 ω ω

2 2

a ’ a† a + a†

’

Hem = +

2 2 2

ω

aa† + a† a .

= (2.67)

2

By using the commutation relation (2.63), this can be written in the equivalent form

1

Hem = ω a† a + . (2.68)

2

The superposition principle (2.59) is enforced by the assumption that the states of

the radiation operator belong to a Hilbert space. The structure of this Hilbert space is

essentially determined by the fact that Hem is a positive operator, i.e. Ψ |Hem | Ψ 0

for any |Ψ . To see this, set |¦ = a |Ψ and use the general rule ¦ |¦ 0 to conclude

that

ω

Ψ |Hem | Ψ = ω Ψ a† a Ψ +

2

ω

= ω ¦ |¦ + 0. (2.69)

2

In particular, this means that all eigenvalues of Hem are nonnegative. Let |φ be an

eigenstate of Hem with eigenvalue W ; then a |φ satis¬es

Hem a |φ = {[Hem , a] + aHem } |φ

= W a |φ + [Hem , a] |φ . (2.70)

The commutator is given by

[Hem , a] = ω a† a, a

= ω a† [a, a] + a† , a a

= ’ ωa , (2.71)

so that

Hem a |φ = (W ’ ω) a |φ . (2.72)

Thus a |φ is also an eigenstate of Hem , but with the reduced eigenvalue (W ’ ω).

Since a lowers the energy by ω, repeating this process would eventually generate

¿

Quantization of cavity modes

states of negative energy. This is inconsistent with the inequality (2.69); therefore, the

Hilbert space of a consistent quantum theory for an oscillator must include a lowest

energy eigenstate |0 satisfying

0| a† = 0 ,

a |0 = 0 , (2.73)

and

ω

Hem |0 = |0 . (2.74)

2

In the case of a mechanical oscillator |0 is the ground state, and a is a lowering

operator. A calculation similar to eqns (2.70) and (2.71) leads to

Hem a† |φ = (W + ω) a† |φ , (2.75)

which shows that a† raises the energy by ω, so a† is a raising operator. The idea

behind this language is that the mechanical oscillator itself is the object of interest.

The energy levels are merely properties of the oscillator, like the energy levels of an

atom.

The equations describing the radiation and mechanical oscillators have the same

form, but there is an important di¬erence in physical interpretation. For the electro-

magnetic ¬eld, it is the quanta of excitation”rather than the radiation oscillators

themselves”that are the main objects of interest. This shift in emphasis incorpo-

rates Einstein™s original proposal that the electromagnetic ¬eld is composed of discrete

quanta. In keeping with this view, it is customary to replace the cumbersome phrase

˜quantum of excitation of the electromagnetic ¬eld™ by the term photon. The in-

tended implication is that photons are physical objects on the same footing as massive

particles. The subtleties associated with treating photons as particles are addressed

in Section 3.6. Since a removes one photon, it is natural to call it the annihilation

operator, and a† , which adds a photon, is naturally called a creation operator. In

this language the ground state of the radiation oscillator is referred to as the vacuum

state, since it contains no photons.

The number operator N = a† a satis¬es the commutation relations

N, a† = a† ,

[N, a] = ’a , (2.76)

so that the a and a† respectively decrease and increase the eigenvalues of N by one.

Since N |0 = 0, this implies that the eigenvalues of N are the the integers 0, 1, 2, . . ..

The eigenvectors of N are called number states, and it is easy to see that N |n =

n |n implies

n