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|n = Zn a† |0 , (2.77)
where Zn is a normalization constant. The Hamiltonian can be written as Hem =
(N + 1/2) ω, so the number states are also energy eigenstates: Hem |n = (n + 1/2)
ω |n . The commutation relations (2.76) can be used to derive the results


1
n |n ’ 1 , and a† |n = n + 1 |n + 1 .
Zn = √ , n |n = δnn , a |n =
n!
(2.78)
Quantization of cavity modes

(1)
The Hilbert space HF for a single mode consists of all linear combinations of
number states, i.e. a typical vector is given by

|Ψ = Cn |n . (2.79)
n=0

(1)
The space HF is called the (single-mode) Fock space. In mathematical jargon”see
(1) (1)
Appendix A.2”HF is said to be spanned by the number states, or HF is said to be
the span of the number states. Since the number states are orthonormal, the expansion
(2.79) can be expressed as

|Ψ = |n n |Ψ . (2.80)
n=0

For any state |φ the expression |φ φ| stands for an operator”see Appendix C.1.2”
that is de¬ned by its action on an arbitrary state |χ :

(|φ φ|) |χ ≡ |φ φ |χ . (2.81)

This shows that |φ φ| is the projection operator onto |φ , and it allows the expansion
(2.80) to be expressed as

|ψ = |n n| |ψ . (2.82)
n=0

The general de¬nition (2.81) leads to

n |) = |n n |n n | = δnn (|n n|) ;
(|n n|) (|n (2.83)

therefore, the (|n n|)s are a family of orthogonal projection operators. According to
eqn (2.82) the projection operators onto the number states satisfy the completeness
relation

|n n| = 1 . (2.84)
n=0

B Vacuum ¬‚uctuations of a single radiation oscillator
A standard argument from quantum mechanics (Bransden and Joachain, 1989, Sec.
5.4) shows that the canonical commutation relations (2.61) for the operators q and p
lead to the uncertainty relation
∆q∆p, (2.85)
2
where the rms deviations ∆q and ∆p are de¬ned by

2 2
Ψ |q 2 | Ψ ’ Ψ |q| Ψ Ψ |p2 | Ψ ’ Ψ |p| Ψ
∆q = , ∆p = , (2.86)

(1)
and |Ψ is any normalized vector in HF . For the vacuum state the relations (2.66)
and (2.73) yield 0 |q| 0 = 0 and 0 |p| 0 = 0, so the uncertainty relation implies that
Quantization of cavity modes

neither 0 q 2 0 nor 0 p2 0 can vanish. For mechanical oscillators this is attributed
to zero-point motion; that is, even in the ground state, random excursions around the
classical equilibrium at Q = P = 0 are required by the uncertainty principle. The
ground state for light is the vacuum state, so the random excursions of the radiation
oscillators are called vacuum ¬‚uctuations. Combining eqn (2.66) with eqn (2.73)
yields the explicit values
ω
0 q2 0 = 0 p2 0 =
, . (2.87)
2ω 2
We note for future reference that the vacuum deviations are ∆q0 = /2ω and ∆p0 =
ω/2, and that these values saturate the inequality (2.85), i.e. ∆q0 ∆p0 = /2. States
with this property are called minimum-uncertainty states, or sometimes minimum-
uncertainty-product states.
The vacuum ¬‚uctuations of the radiation oscillator also explain the fact that the
energy eigenvalue for the vacuum is ω/2 while the classical energy minimum is Uem =
0. Inserting eqn (2.87) into the original expression eqn (2.65) for the Hamiltonian yields
0 |Hem | 0 = ω/2. The discrepancy between the quantum and classical minimum
energies is called the zero-point energy; it is required by the uncertainty principle for
the radiation oscillator. Since energy is only de¬ned up to an additive constant, it would
be permissible”although apparently unnatural”to replace the classical expression
(2.64) by
12 ω
U= p + ω2 q2 ’ . (2.88)
2 2
Carrying out the substitution (2.66) on this expression yields the Hamiltonian

Hem = ωa† a . (2.89)

With this convention the vacuum energy vanishes for the quantum theory, but the
discrepancy between the quantum and classical minimum energies is unchanged. The
same thing can be accomplished directly in the quantum theory by simply subtracting
the zero-point energy from eqn (2.68). Changes of this kind are always permitted, since
only di¬erences of energy eigenvalues are physically meaningful.

C The multi-mode Fock space
Since the classical radiation oscillators in the cavity are mutually independent, the
quantization rule is given by eqns (2.60)“(2.63), and the only real di¬culties stem from
the fact that there are in¬nitely many modes. For each mode, the number operator
Nκ = a† aκ is evidently positive and satis¬es
κ

[Nκ , a» ] = ’δκ» aκ , (2.90)

Nκ , a† = δκ» a† . (2.91)
κ
»

Combining eqn (2.90) with the positivity of Nκ and applying the argument used for
the single-mode Hamiltonian in Section 2.1.2-A leads to the conclusion that there must
be a (multimode) vacuum state |0 satisfying
Quantization of cavity modes

aκ |0 = 0 for every mode-index κ . (2.92)

Since number operators for di¬erent modes commute, it is possible to ¬nd vectors |n
that are simultaneous eigenstates of all the mode number operators:

Nκ |n = nκ |n for all κ ,
(2.93)
n = {nκ for all κ} .

According to the single-mode results (2.77) and (2.78) the many-mode number states
are given by

a†
κ

|n = |0 . (2.94)
nκ !
κ

The total number operator is

a† aκ ,
N= (2.95)
κ
κ

and
N |n = |n .
nκ (2.96)
κ

The Hilbert space HF spanned by the number states |n is called the (multimode)
Fock space.
It is instructive to consider the simplest number states, i.e. those containing exactly
one photon. If κ and » are the labels for two distinct modes, then eqn (2.96) tells us
that |1κ = a† |0 and |1» = a† |0 are both one-photon states. The same equation
κ »
tells us that the superposition
1 1 1
|ψ = √ |1κ + √ |1» = √ a† + a† |0 (2.97)
κ »
2 2 2
is also a one-photon state; in fact, every state of the form

ξκ a† |0
|ξ = (2.98)
κ
κ

is a one-photon state. There is a physical lesson to be drawn from this algebraic
exercise: it is a mistake to assume that photons are necessarily associated with a single
classical mode. Generalizing this to a superposition of modes which form a classical
wave packet, we see that a single-photon wave packet state (that is, a wave packet
that contains exactly one photon) is perfectly permissible.
According to eqn (2.94) any number of photons can occupy a single mode. Further-
more the commutation relations (2.63) guarantee that the generic state a† 1 · · · a† n |0
κ κ
is symmetric under any permutation of the mode labels κ1 , . . . , κn . These are de¬n-
ing properties of objects satisfying Bose statistics (Bransden and Joachain, 1989, Sec.
10.2), so eqns (2.63) are called Bose commutation relations and photons are said to
be bosons.
Normal ordering and zero-point energy

D Field operators
In the Schr¨dinger picture, the operators for the electric and magnetic ¬elds are”by
o
de¬nition”time-independent. They can be expressed in terms of the time-independent
operators pκ and qκ by ¬rst using the classical expansions (2.40) and (2.41) to write the
initial classical ¬elds E (r, 0) and B (r, 0) in terms of the initial displacements Qκ (0)
and momenta Pκ (0) of the radiation oscillators. Setting (Qκ , Pκ ) = (Qκ (0) , Pκ (0)),
and applying the quantization conjecture (2.60) to these results leads to
1
E (r) = ’ √ pκ E κ (r) , (2.99)
0 κ

1
kκ qκ Bκ (r) .
B (r) = √ (2.100)
0 κ
For most applications it is better to express the ¬elds in terms of the operators aκ and
a† by using eqn (2.66) for each mode:
κ

ωκ
aκ ’ a† E κ (r) ,
E (r) = i (2.101)
κ
20
κ

µ0 ωκ
aκ + a† Bκ (r) .
B (r) = (2.102)
κ
2
κ
The corresponding expansions for the vector potential in the radiation gauge are
1
qκ E κ (r)
A (r) =
0
κ

aκ + a† E κ (r) .
= (2.103)
κ
2 0 ωκ
κ

2.2 Normal ordering and zero-point energy
In the absence of interactions between the independent modes, the energy is additive;
therefore, the Hamiltonian is the sum of the Hamiltonians for the individual modes.
If we use eqn (2.68) for the single-mode Hamiltonians, the result is
ωκ
ω κ a† aκ +
Hem = . (2.104)
κ
2
κ

The previously innocuous zero-point energies for each mode have now become a serious
annoyance, since the sum over all modes is in¬nite. Fortunately there is an easy way
out of this di¬culty. We can simply use the alternate form (2.89) which gives

ω κ a† aκ .
Hem = (2.105)
κ
κ

With this choice for the single-mode Hamiltonians the vacuum energy is reduced from
in¬nity to zero.
Quantization of cavity modes

It is instructive to look at this problem in a di¬erent way by using the equivalent
form eqn (2.67), instead of eqn (2.68), to get
ωκ †
aκ a κ + aκ a † .
Hem = (2.106)
κ
2
κ

Now the zero-point energy can be eliminated by the simple expedient of reversing the
order of the operators in the second term. This replaces eqn (2.106) by eqn (2.105). In
other words, subtracting the vacuum expectation value of the energy is equivalent to
reordering the operator products so that in each term the annihilation operator is to
the right of the creation operator. This is called normal ordering, while the original
order in eqn (2.106) is called symmetrical ordering.
We are allowed to consider such a step because there is a fundamental ambiguity
involved in replacing products of commuting classical variables by products of non-
commuting operators. This problem does not appear in quantizing the classical energy
expression in eqn (2.64), since products of qκ with pκ do not occur. This happy cir-
cumstance is a fortuitous result of the choice of classical variables. If we had instead
chosen to use the variables ±κ de¬ned by eqn (2.45), the quantization conjecture would
be ±κ ’ aκ and ±— ’ a† . This does produce an ordering ambiguity in quantizing eqn
κ κ
(2.51), since ±κ ±— , (±— ±κ + ±κ ±— ) /2, and ±— ±κ are identical in the classical theory,
κ κ κ κ
but di¬erent after quantization. The last two forms lead respectively to the expressions
(2.106) and (2.105) for the Hamiltonian. Thus the presence or absence of the zero-point
energy is determined by the choice of ordering of the noncommuting operators.
It is useful to extend the idea of normal ordering to any product of operators
X1 · · · Xn , where each Xi is either a creation or an annihilation operator. The normal-
ordered product is de¬ned by

: X1 · · · Xn : = X 1 · · · Xn , (2.107)

where (1 , . . . , n ) is any ordering (permutation) of (1, . . . , n) that arranges all of the
annihilation operators to the right of all the creation operators. The commutation
relations are ignored when carrying out the reordering. More generally, let Z be
a linear combination of distinct products X1 · · · Xn ; then : Z : is the same linear
combination of the normal-ordered products : X1 · · · Xn : . The vacuum expectation
value of a normal-ordered product evidently vanishes, but it is not generally true that
Z = : Z : + 0 |Z| 0 .

2.3 States in quantum theory
In classical mechanics, the coordinate q and momentum p of a particle can be precisely
speci¬ed. Therefore, in classical physics the state of maximum information for a system
of N particles is a point q, p = (q1 , p1 , . . . , qN , pN ) in the mechanical phase space.
For large values of N , specifying a point in the phase space is a practical impossibility,
so it is necessary to use classical statistical mechanics”which describes the N -body
system by a probability distribution f q, p ”instead. The point to bear in mind here
is that this probability distribution is an admission of ignorance. No experimentalist
can possibly acquire enough information to determine a particular value of q, p .
States in quantum theory

In quantum theory, the uncertainty principle prohibits simultaneous determination
of the coordinates and momenta of a particle, but the notions of states of maximum
and less-than-maximum information can still be de¬ned.

2.3.1 Pure states
In the standard interpretation of quantum theory, the vectors in the Hilbert space de-
scribing a physical system”e.g. general linear combinations of number states in Fock
space”provide the most detailed description of the state of the system that is consis-
tent with the uncertainty principle. These quantum states of maximum information
are called pure states (Bransden and Joachain, 1989, Chap. 14). From this point of
view the random ¬‚uctuations imposed by the uncertainty principle are intrinsic; they
are not the result of ignorance of the values of some underlying variables.
For any quantum system the average of many measurements of an observable X
on a collection of identical physical systems, all described by the same vector |Ψ , is
given by the expectation value Ψ |X| Ψ . The evolution of a pure state is described
by the Schr¨dinger equation
o

|Ψ (t) = H |Ψ (t) ,
i (2.108)
‚t
where H is the Hamiltonian.

2.3.2 Mixed states
In the absence of maximum information, the system is said to be in a mixed state.
In this situation there is insu¬cient information to decide which pure state describes
the system. Just as for classical statistical mechanics, it is then necessary to assign a
probability to each possible pure state. These probabilities, which represent ignorance
of which pure state should be used, are consequently classical in character.
As a simple example, suppose that there is only su¬cient information to say that
each member of a collection of identically prepared systems is described by one or the
other of two pure states, |Ψ1 or |Ψ2 . For a system described by |Ψe (e = 1, 2), the
average value for measurements of X is the quantum expectation value Ψe |X| Ψe .
The overall average of measurements of X is therefore

X = P1 Ψ1 |X| Ψ1 + P2 Ψ2 |X| Ψ2 , (2.109)

where Pe is the fraction of the systems described by |Ψe , and P1 + P2 = 1.
The average in eqn (2.109) is quite di¬erent from the average of many measure-
ments on systems all described by the superposition state |Ψ = C1 |Ψ1 + C2 |Ψ2 . In
that case the average is

2 2
Ψ |X| Ψ = |C1 | Ψ1 |X| Ψ1 + |C2 | Ψ2 |X| Ψ2 + 2 Re [C1 C2 Ψ1 |X| Ψ2 ] ,
(2.110)
which contains an interference term missing from eqn (2.109). The two results (2.109)

and (2.110) only agree when |Ce |2 = Pe and Re [C1 C2 Ψ1 |X| Ψ2 ] = 0. The latter

condition can be satis¬ed if C1 C2 Ψ1 |X| Ψ2 is pure imaginary or if Ψ1 |X| Ψ2 = 0.
Since it is always possible to choose another observable X for which neither of these
¼ Quantization of cavity modes

conditions is satis¬ed, it is clear that the mixed state and the superposition state
describe very di¬erent physical situations.

A The density operator
In general, a mixed state is de¬ned by a collection, usually called an ensemble, of
normalized pure states {|Ψe }, where the label e may be discrete or continuous. For
simplicity we only consider the discrete case here: the continuum case merely involves
replacing sums by integrals with suitable weighting functions. For the discrete case, a
probability distribution on the ensemble is a set of real numbers {Pe } that satisfy
the conditions
0 Pe 1 , (2.111)

Pe = 1 . (2.112)
e

The ensemble may be ¬nite or in¬nite, and the vectors need not be mutually orthog-
onal.
The average of repeated measurements of an observable X is represented by the
ensemble average of the quantum expectation values,

Pe Ψe (t) |X| Ψe (t) ,
X (t) = (2.113)
e

where |Ψe (t) is the solution of the Schr¨dinger equation with initial value |Ψe (0) =
o
|Ψe . It is instructive to rewrite this result by using the number-state basis {|n } for
Fock space to get

Ψe (t) |X| Ψe (t) = Ψe (t) |n n |X| m m |Ψe (t) , (2.114)
n m


and
n |X| m Pe m |Ψe (t) Ψe (t) |n
X (t) = . (2.115)
n m e

By applying the general de¬nition (2.81) to the operator |Ψe (t) Ψe (t)|, it is easy
to see that the quantity in square brackets is the matrix element m |ρ (t)| n of the
density operator:
Pe |Ψe (t) Ψe (t)| .
ρ (t) = (2.116)
e

With this result in hand, eqn (2.115) becomes

m |ρ (t)| n n |X| m

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