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for the Fourier transforms of the susceptibilities, but there is a di¬erent”
(n)
and frequently used”convention in which χij1 j2 ···jn (ν1 , ν2 , . . . , νn ) is replaced by
(n)
χij1 j2 ···jn (’ν0 , ν1 , ν2 , . . . , νn ), with the understanding that the sum of the frequency
arguments is zero (Boyd, 1992, Sec. 1.5). This is an example of the notational schisms
that are common in this ¬eld. The nth-order frequency-domain susceptibility tensor
is symmetrical under simultaneous permutations of νp and jp , and the reality of the
time-domain susceptibility imposes the conditions
(n)— (n)
χij1 j2 ···jn (ν1 , . . . , νn ) = χij1 j2 ···jn (’ν1 , . . . , ’νn ) (B.157)

in the frequency domain. For the transparent media normally considered, the Fourier
(n)
transform χij1 j2 ···jn (ν1 , . . . , νn ) is also real, and eqn (B.157) becomes
(n) (n)
χij1 j2 ···jn (ν1 , . . . , νn ) = χij1 j2 ···jn (’ν1 , . . . , ’νn ) . (B.158)

The properties listed above give no information regarding what happens if the ¬rst
index i is interchanged with one of the jp s. For transparent media, the explicit quan-
tum perturbation calculation of the susceptibilities provides the additional symmetry
condition (Boyd, 1992, Sec. 3.2)
n
(n) (n)
ν1 , . . . , ’
χij1 ···jp ···jn (ν1 , . . . , νp , . . . , νn ) = χjp j1 j2 ···i···jn νk , . . . , νn . (B.159)
k=1
Appendix C
Quantum theory

Modern quantum theory originated with the independent inventions of matrix mechan-
ics by Heisenberg and wave mechanics by Schr¨dinger. It was essentially completed
o
by Schr¨dinger™s proof that the two formulations are equivalent and Born™s interpre-
o
tation of the wave function as a probability amplitude. The intuitive appeal of wave
mechanics, at least for situations involving a single particle, explains its universal use
in introductory courses on quantum theory. This approach does, however, have certain
disadvantages. One is that the intuitive simplicity of wave mechanics is largely lost
when it is applied to many-particle systems. For our purposes, a more serious objection
is that there are no wave functions for photons.
A more satisfactory approach is based on the fact that interference phenomena
are observed for all microscopic systems. For example, the two-slit experiment can
be performed with material particles to observe interference fringes. A comparison to
macroscopic wave phenomena suggests that the mathematical description of the states
of a system should satisfy the superposition principle, i.e. every linear combination
of states is also a state. In mathematical terms this means that the states are elements
of a vector space, and the Born interpretation”to be explained below”requires the
vector space to be a Hilbert space.

C.1 Dirac™s bra and ket notation
In Appendix A.3 Hilbert spaces are described with the standard notation used in
mathematics and in many textbooks on quantum theory. In the main text, we employ
an alternative notation introduced by Dirac (1958), in which a vector in a Hilbert space
H is represented by the symbol |ψ . In this notation | · represents a generic ket vector
and ψ is a label that distinguishes one vector from another. Linear combinations of
two kets, |ψ and |φ , are written as ± |ψ + β |φ , and scalars, like ± and β, are called
c-numbers.
In the Dirac notation, a bra vector F | represents a rule that assigns a complex
number, denoted by F |ψ , to every ket vector |ψ . This rule is linear, i.e. if |ψ =
± |χ + β |φ , then
F |ψ = ± F |χ + β F |φ . (C.1)
The Hilbert-space inner product (φ, ψ) is an example of such a rule, so for each ket
vector |φ there is a corresponding bra vector φ| (called the adjoint vector) de¬ned
by
φ |ψ = (φ, ψ) for all ψ . (C.2)
With this understanding, we will use • |ψ from now on to denote the inner product.
½
Dirac™s bra and ket notation

The linearity of the rule (C.1) guarantees that the set of bra vectors is in fact a
vector space. The o¬cial jargon”explained in Appendix A.6.1”is that the bra vectors
form the dual space H of linear functionals on H. The de¬nition (C.2) of the adjoint
vectors shows that the Hilbert space H of physical states is isomorphic to a subspace
of H .
The Hilbert spaces relevant for quantum theory are always separable; that is, every
ket |ψ can be expanded as

|ψ = |φn φn |ψ , (C.3)
n

where {|φn , n = 1, 2, . . .} is an orthonormal basis for H.

C.1.1 Examples
A Two-level system
The states of a two-level system, e.g. a spin-1/2 particle, are usually represented by
two-component column vectors that refer to a given basis, e.g. eigenstates of σz . The
relation between this concrete description and the Dirac notation is

ψ1 — —
• |ψ = (•, ψ) = •— ψ1 + •— ψ2 .
|ψ ∼ ψ| ∼ (ψ1 , ψ2 ) ,
, (C.4)
1 2
ψ2

The symbol ˜∼™ is used instead of ˜=™ because the values of the components ψ1 and
ψ2 depend on the particular choice of basis in the concrete space C2 . A di¬erent basis
choice would represent the same ket vector |ψ by a di¬erent pair of components ψ1 ,
ψ2 . An example of an orthonormal basis is

1 0
B= |1 ∼ , |2 ∼ , (C.5)
0 1

so the components are given by ψ1 = 1 |ψ and ψ2 = 2 |ψ , and

|ψ = ψ1 |1 + ψ2 |2 . (C.6)

This relation is invariant under a change of basis in C2 , since the vectors |1 and |2
would also be transformed. Every bra vector (linear functional) on C2 is de¬ned by
taking the inner product with some ¬xed vector in C2 , so the space of bra vectors
(the dual space) is isomorphic to the space itself, i.e. H = H. This is true for any
¬nite-dimensional Hilbert space.

B Spinless particle in three dimensions
As a second example, consider the familiar description of a spinless particle by a
square-integrable wave function ψ(r). The square-integrability condition is

d3 r |ψ (r)|2 < ∞ , (C.7)
’∞
¾ Quantum theory

and the set of square-integrable functions is called L2 R3 . The relation between the
abstract and concrete descriptions is

ψ| ∼ ψ — (r) , d3 r•— (r) ψ (r) ,
|ψ ∼ ψ (r) , • |ψ = (C.8)

where the vector operations are de¬ned point-wise:

± |ψ + β |• ∼ ±ψ (r) + β• (r) . (C.9)

For in¬nite-dimensional Hilbert spaces, such as H = L2 R3 , there are bra vectors
that are not adjoints of any vector in the space. In other words, the dual space H is
larger than the space H. For example, the delta function δ (r ’ r0 ) is not the adjoint
of any vector in L2 R3 , but it does de¬ne a bra vector r0 | by

r0 |ψ = d3 rδ (r ’ r0 ) ψ (r) = ψ (r0 ) . (C.10)

This establishes the relation ψ (r) = r |ψ between the concrete and abstract descrip-
tions.
Although the bra vector r0 | is not the adjoint of any proper ket vector (nor-
malizable wave function) in L2 R3 , it is common practice to de¬ne an improper
ket vector |r0 by the rule ψ |r0 = ψ — (r0 ) for all ψ ∈ H. The position opera-
tor r is de¬ned by rψ (r) = rψ (r), and |r0 is an improper eigenvector of r”i.e.
r |r0 = r0 |r0 ”by virtue of

= r0 ψ — (r0 ) = r0 ψ |r0 .
ψ |r| r0 = r0 |r| ψ (C.11)

In the same way, there is no proper eigenvector of the momentum operator p, but
there is an improper eigenvector |p0 , i.e. p |p0 = p0 |p0 , associated with the bra
vector p0 | de¬ned by
d3 re’ip0 ·r/ ψ (r) .
p0 |ψ = (C.12)

C.1.2 Linear operators
The action of a linear operator A is denoted by A |ψ , and the complex number
ψ |A| • is the matrix element of the operator A for the pair of vectors |ψ and
|• . The operator A is uniquely determined by any of the sets of matrix elements

{ φn |A| φm } for all |φn , |φm in a basis B , (C.13)
{ ψ |A| • } for all |ψ , |• in H , (C.14)
{ ψ |A| ψ } for all |ψ in H . (C.15)

The operator T•χ , de¬ned by the rule

T•χ |ψ = |• χ |ψ for all |ψ , (C.16)

is usually written as |• χ|. The product of two such operators therefore acts by
¿
Physical interpretation

T•χ Tβξ |ψ = T•χ |β ξ |ψ = |• χ |β ξ |ψ . (C.17)

This holds for all states |ψ , so the product rule is

T•χ Tβξ = χ |β T•ξ . (C.18)

The operator T•• = |• •| is therefore a projection operator, provided that |• is
normalized.
Let {|φn } be an orthonormal basis for a subspace W ‚ H; then the projection
operators Pn = |φn φn | are orthogonal, i.e. Pn Pm = δnm . Every vector |ψ in W has
the unique expansion

|ψ = |φn φn |ψ = Pn |ψ , (C.19)
n n

so the operator
|φn φn |
PW = Pn = (C.20)
n n

acts as the identity for vectors in W. On the other hand, every vector |χ in the
orthogonal complement W⊥ is annihilated by PW , i.e. PW |χ = 0, so PW is the
projection operator onto W. When W = H the projection PH is the identity operator
and we get
|φn φn | = I , (C.21)
n

which is called the completeness relation, or a resolution of the identity into
the projection operators Pn = |φn φn | .
If B = {|•1 , |•2 , . . .} is an orthonormal basis, then the trace of A is de¬ned by

φn |A| φn .
Tr (A) = (C.22)
n

The value of Tr (A) is the same for all choices of orthonormal basis, and

Tr (AB) = Tr (BA) . (C.23)

The last property is called cyclic invariance, since it implies

Tr (A1 A2 · · · An ) = Tr (An A1 A2 · · · An’1 ) . (C.24)

C.2 Physical interpretation
The mathematical formalism is connected to experiment by the following assump-
tions.
(1) The states of maximum information, called pure states, are vectors in a Hilbert
space H.
(2) Each observable quantity is represented by a Hermitian operator A, and the value
obtained in a measurement is always one of the eigenvalues an of A. Hermitian
operators are, therefore, often called observables.
Quantum theory

(3) If the system is prepared in the state |ψ , then the probability that a measurement
of A yields the value an is | φn |ψ |2 , where A |φn = an |φn . This is the Born
interpretation (Born, 1926). After the measurement is performed, the system is
described by the eigenvector |φn . This is the infamous reduction of the wave
packet.
(a) This description implicitly assumes that the eigenvalue an is nondegenerate. In
the more typical case of an eigenvalue with degeneracy d > 1, the probability
for ¬nding an is
d
| φnk |ψ |2 , (C.25)
k=1

where {|φnk , k = 1, . . . , d} is an orthonormal basis for the an -eigenspace. The
corresponding projection operator is
d
|φnk φnk | .
Pn = (C.26)
k=1

(b) Von Neumann™s projection postulate (von Neumann, 1955) states that
the probability of ¬nding an is
d
2
ψ |Pn | ψ = | φnk |ψ | , (C.27)
k=1

and”for ψ |Pn | ψ = 0”the ¬nal state after the measurement is
1
|ψ¬n = Pn |ψ . (C.28)
ψ |Pn | ψ
(c) An alternative way of dealing with degeneracies is to replace the single observ-
able A by a set of observables {A1 , A2 , . . . , AN } with the following properties.
(i) The operators are mutually commutative, i.e. [Ai , Aj ] = 0.
(ii) A vector |φ that is a simultaneous eigenvector of all the Ai s” i.e. Ai |φ =
ai |φ for i = 1, . . . , N ”is uniquely determined (up to an overall phase
factor).
A set {A1 , A2 , . . . , AN } with these properties is called a complete set of
commuting observables (CSCO). A simultaneous measurement of the ob-
servables in the CSCO leaves the system in a state that is unique except for
an overall phase factor.
(4) The average of many measurements of A performed on identical systems prepared
in the state |ψ is the expectation value ψ |A| ψ .
(5) There is a special Hermitian operator, the Hamiltonian H, which describes
the time evolution”often called time translation”of the system through the
Schr¨dinger equation
o

|ψ (t) = H (t) |ψ (t) .
i (C.29)
‚t
Useful results for operators

The explicit time dependence of the Hamiltonian can only occur in the presence
of external classical forces.

C.3 Useful results for operators
C.3.1 Pauli matrices
Consider linear operators on the space C2 . It is easy to see that every operator is
represented by a 2 — 2 matrix, so it is determined by four complex numbers. The
Pauli matrices, de¬ned by

0 ’i
01 1 0
σx = σ1 = , σy = σ2 = , σz = σ3 = , (C.30)
’1
10 i0 0

are particularly important. They satisfy the commutation relations

[σi , σj ] = 2i ijk σk , (C.31)

where ijk is the alternating tensor de¬ned by eqn (A.3), and the anticommutation
relations
[σi, σj ]+ = σi σj + σj σi = 2δij (i, j = x, y, z) , (C.32)
which combine to yield
σi σj = ijk σk + δij . (C.33)
It is often useful to use the so-called circular basis {σz , σ± = (σx ± iσy ) /2} with the
commutation relations

[σz , σ± ] = ±2σ± , [σ+ , σ’ ] = σz , (C.34)

and the anticommutation relations

[σ± , σ± ]+ = 0 , [σ± , σ“ ]+ = 1 , [σz , σ± ]+ = 0 . (C.35)

These fundamental relations yield the useful identities
1
σ+ σ’ = (1 + σz ) , (C.36)
2
1
σ’ σ+ = (1 ’ σz ) , (C.37)
2
σz σ± = ±σ± = ’σ± σz . (C.38)
The three Pauli matrices, together with the identity matrix, are linearly indepen-
dent and therefore constitute a complete set for the expansion of all 2 — 2 matrices.
Thus every 2 — 2 matrix A has the representation

A = a0 σ0 + ai σi , (C.39)

where σ0 is the identity matrix. These properties, together with the observation that
Tr (σi ) = 0, yield
Quantum theory

1
a0 = Tr (A) ,
2 (C.40)
1
aj = Tr (Aσj ) .
2
2 2
Writing ai σi = a · σ and using the properties given above yields (a · σ) = |a| , and
this in turn provides the useful identities (Cohen-Tannoudji et al., 1977b, Complement
A-IX)

ei±u·σ = cos (±) + i sin (±) u · σ , (C.41)
eβu·σ = cosh (β) + sinh (β) u · σ , (C.42)

where ± and β are real constants and u is a real unit vector.

C.3.2 The operator binomial theorem
For c-numbers x and y the binomial theorem is
n
n!
n
xn’p y p ,
(x + y) = (C.43)
p! (n ’ p)!
p=0


but this depends on the fact that c-numbers commute. For noncommuting operators X
and Y the quantity (X + Y )n is to be evaluated by multiplying together the n factors
X + Y . Consider the terms of order (n ’ p, p) in this expansion, i.e. those in which X
occurs n ’ p times and Y occurs p times. Since each of these terms is the product of n
factors, there are a total of n! orderings. The orderings that di¬er only by exchanging
Xs with Xs or Y s with Y s are identical, and the number of these terms is precisely
the binomial coe¬cient n!/p! (n ’ p)!; therefore,
n
n!
n
S X n’m Y m ,
(X + Y ) = (C.44)
p! (n ’ p)!
m=0


where S [X n’m Y m ] is the average of the terms with (n ’ m) Xs and m Y s arranged
in all possible orders. This is called the symmetrical or Weyl product.
For (n, 0) or (0, n) one has simply S [X n ] = X n or S [Y n ] = Y n . Examples of
mixed powers are

1
S [XY ] = (XY + Y X) ,
2
1
S X 2Y = X 2 Y + XY X + Y X 2 ,
3
1
S X 2Y 2 = X 2 Y 2 + XY 2 X + XY XY + Y 2 X 2 + Y X 2 Y + Y XY X ,
6
.
.
. (C.45)
Useful results for operators

C.3.3 Commutator identities
The Leibnitz rule
[A, BC] = A [B, C] + [A, B] C (C.46)
and the Jacobi identity

[[A, B] , C] + [[C, A] , B] + [[B, C] , A] = 0 (C.47)

are both readily veri¬ed by direct use of the de¬nition [A, B] = AB ’ BA. The useful
identity ⎛ ⎞ ⎛ ⎞
p’1
n n
⎝ Bj ⎠ [A, Bp ] ⎝ Bk ⎠
[A, B1 B2 · · · Bn ] = (C.48)
p=1 j=1 k=p+1

can be established by an induction argument, combined with the convention that an
empty product has the value unity. In the special case that each single commutator
[A, Bp ] commutes with the remaining Bj s, this becomes
⎛ ⎞
n n
[A, Bp ] ⎝ Bj ⎠ .
[A, B1 B2 · · · Bn ] = (C.49)
p=1 j=p=1

C.3.4 Operator expansion theorems
Theorem C.1 Let X and Y be operators acting on a Hilbert space H. Then

κn
’κX (n)
κX
e Ye = [X, Y ] , (C.50)
n!
n=0

(n) (0)
where the iterated commutator [X, Y ] is de¬ned by the initial value [X, Y ] =Y
and the recursion relations
(n+1) (n)
[X, Y ] = X, [X, Y ] for n 0. (C.51)



Proof Let Y (κ) ≡ eκX Y e’κX ; then dY (κ) /dκ = [X, Y (κ)]. Iterating this result
implies
dn+1 Y (κ) dn Y (κ)
= X, , (C.52)
dκn+1 dκn
and eqn (C.50) follows by a Taylor series expansion around κ = 0.
In the special case that the commutator [X, Y ] commutes with X, the series ter-
minates so that
eκX Y e’κX = Y + κ [X, Y ] , (C.53)
i.e. X generates translations of Y . An important example is a canonically conjugate
pair: X = p, Y = q, with [q, p] = i . Choosing κ = iu/ , where u is a c-number, gives
the familiar quantum mechanics result
Quantum theory


Tu qTu = q + u , (C.54)

where the unitary operator
Tu = e’iup/ (C.55)
evidently generates translations in the position. For any well-behaved operator function
F (q), e.g. one that has a Taylor series expansion, the last result generalizes to

Tu F (q) Tu = F (q + u) . (C.56)

For in¬nitesimal values of u, expanding both sides leads to

‚F (q)
[p, F (q)] = ’i . (C.57)
‚q

To see the action of Tu on a state vector, rewrite eqn (C.54) as qTu = Tu (q + u) and
apply this to an eigenvector |Q of q to get

qTu |Q = Tu (q + u) |Q = Tu (Q + u) |Q = (Q + u) Tu |Q ; (C.58)

in other words,
Tu |Q = |Q + u . (C.59)
Thus for any state |Ψ ,
Q |Tu | Ψ = Q + u |Ψ , (C.60)
or in more familiar notation

(Tu Ψ) (Q) = Ψ (Q + u) . (C.61)

It is also useful to consider the opposite assignment X = q, Y = p, κ = ’iv/ , which
produces
e’ivq/ qeivq/ = p + v , (C.62)
and shows that the position operator generates translations in the momentum.
Another important special case is [X, Y ] = ±Y , where ± is a c-number. Putting
this into the de¬nition (C.51) gives

[X, Y ](n) = ±n Y , (C.63)

so that eqn (C.50) becomes
eκX Y e’κX = e±κ Y . (C.64)
As an example, let X = a† a, Y = a, and κ = iθ, where a is the lowering operator for a
harmonic oscillator. The commutation relation a, a† = 1 yields [X, Y ] = a† a, a =
’a, so
† †
eiθa a ae’iθa = e’iθ a .
a
(C.65)
Useful results for operators

C.3.5 Campbell“Baker“Hausdor¬ theorem
Theorem C.2 Let X and Y be operators such that [X, Y ] commutes with both X and
Y . Then
1
eX eY = eX+Y e 2 [X,Y ] . (C.66)


Proof See Peres (1995, Sec. 10-7).
Two important special cases are needed in the text. The ¬rst is de¬ned by setting
X = ’ivq, Y = ’iup, which leads to

e’i(up+vq) = ei uv/2 ’ivq ’iup
e e . (C.67)

Interchanging the de¬nitions of X and Y produces

e’i(up+vq) = e’i uv/2 ’iup ’ivq
e e . (C.68)

The second example is X = κa† , Y = ’κ— a, which gives

’κ— a /2 κa† ’κ— a
= e’|κ|
2
eκa e e . (C.69)

Interchanging X and Y yields the alternative identity

’κ— a /2 ’κ— a κa†
= e|κ|
2
eκa e e . (C.70)

C.3.6 Functions of operators
Let X be a Hermitian operator and f (u) be a real-valued function of the real variable
u. A vector |ψ is uniquely represented by the expansion

|ψ = |φn φn |ψ , (C.71)
n

where the |φn s are the basis of eigenvectors of X, i.e. X |φn = xn |φn . Then f (X)
is de¬ned by
f (X) |φn = f (xn ) |φn for all n , (C.72)
so that
f (X) |ψ = |φn f (xn ) φn |ψ . (C.73)
n

If the function f (u) has a Taylor series expansion around the value u = u0 ,

n
fn (u ’ u0 ) ,
f (u) = (C.74)
n=0

then an alternative de¬nition of f (X) is

fn (X ’ u0 )n .
f (X) = (C.75)
n=0
¼ Quantum theory

C.3.7 Generalized uncertainty relation
Choose a ¬xed vector |ψ and a pair of normal operators C and D, i.e. C, C † =
D, D† = 0. Use the shorthand notation C = ψ |C| ψ , D = ψ |D| ψ to de¬ne
the ¬‚uctuation operators δC = C’ C and δD = D’ D . Note that [C, D] = [δC, δD].
The expectation value of the commutator is

[C, D] = [δC, δD] = δCδD ’ δDδC ; (C.76)

consequently,
| [C, D] | | δCδD | + | δDδC | . (C.77)
Next set ψ |δCδD| ψ = φ |χ , where |φ = δC † |ψ and |χ = δD |ψ . The Cauchy“
Schwarz inequality (A.9) yields

δCδC † δD† δD .
| φ |χ | φ |φ χ |χ = (C.78)

With the de¬nitions of the rms deviations

∆C 2 = δC † δC = δCδC † ,
(C.79)
∆D2 = δD† δD = δDδD† ,

we ¬nd
δCδC † δD† δD = ∆C ∆D .
| δCδD | = | φ |χ | (C.80)
Interchanging C and D gives

δDδD† δC † δC = ∆C ∆D ,
| δDδC | (C.81)

and putting everything together yields the generalized uncertainty relation
1
| [C, D] | (C.82)
∆C ∆D
2
for any pair of normal operators.

C.4 Canonical commutation relations
Hermitian operators Q and P that satisfy the canonical commutation relation
[Q, P ] = i are said to be canonically conjugate. Applying eqn (C.82) to this case
yields the canonical uncertainty relation

∆Q ∆P /2 . (C.83)

A state for which equality is attained, i.e.

∆Q ∆P = /2 , (C.84)

is called a minimum-uncertainty state or minimum-uncertainty wave packet.
½
Canonical commutation relations

The creation and annihilation operators de¬ned in Section 2.1.2 satisfy the alter-
native form
a M , a† = δMM , [aM , aM ] = 0 (C.85)
M

of the canonical commutation relations. We ¬rst show that these relations are preserved
by any unitary transformation. Let U be a unitary operator and de¬ne new operators

b M = U aM U † ; (C.86)

then
bM , b† = U aM U † , U a† U † = δMM ,
M M
(C.87)
† †
[bM , bM ] = U aM U , U aM U = 0.
The converse statement is also true. If the operators bM satisfy

bM , b† = δMM , [bM , bM ] = 0 , (C.88)
M


then there is a unitary transformation U which relates the bM s and aM s by eqn (C.86).
The proof of this claim depends on the argument in Section 2.1.2-A showing that a
Hilbert space in which eqn (C.85) holds is spanned by the number states, which we
will now call |n; a , satisfying

a† aM |n; a = nM |n; a , n = (n1 , n2 , . . .) . (C.89)
M

This argument applies equally well to the bM s, so there is also a basis of states, |n; b ,
satisfying
b† bM |n; b = nM |n; b . (C.90)
M

It is easy to check that the operator U , de¬ned by

|n; b n; a| ,
U= (C.91)
n


is unitary, and that

U aM U † = |m ; b m ; a |aM | m; a m; b|
m
m

|m ’ 1M ; b
= mM m; b| , (C.92)
m


where m ’ 1M signi¬es (m1 , m2 , . . . , mM ’ 1, . . .). Calculating the general matrix el-
ement of U aM U † in the |n; b basis yields

n; b U aM U † n ; b = δn,n ’1M nM = n; b |bM | n ; b ; (C.93)

therefore, this U satis¬es eqn (C.86).
¾ Quantum theory

C.5 Angular momentum in quantum mechanics
In classical mechanics, the angular momentum of a particle (relative to the origin of
coordinates) is r—p, where p is the momentum. In quantum mechanics (Bransden and
Joachain, 1989, Chap. 6) this becomes the operator L = r — (’i ∇), which satis¬es
the angular momentum commutation relations

[Li , Lj ] = i ijk Lk . (C.94)

Because of its relation to the classical angular momentum used to describe orbits, L
is called the orbital angular momentum. This operator is also related to spatial
rotations, r ’ r = R (n, ‘) r, where R (n, ‘) is a 3 — 3 orthogonal matrix (RT R =
RRT = 1), n is a unit vector de¬ning the axis of rotation, and ‘ is the angle of rotation
around the axis. For small ‘ one can show that

δrj = rj ’ rj = δri = ‘ ijk nj rk . (C.95)

By de¬nition, a vector V transforms like r under rotations.
A scalar wave function ψ (r) transforms according to ψ (r) = U (n, ‘) ψ (r), where
the unitary operator U (n, ‘) is given by
i
U (n, ‘) = exp ’ ‘n · L . (C.96)

Thus L is the generator of spatial rotations.
The corresponding transformation for an operator O is O = U (R) OU † (R). Ex-
panding to ¬rst order for small ‘ gives the in¬nitesimal transformation
i
δO = O ’ O = ‘ [O, n · L] . (C.97)

Combining eqn (C.95) with eqn (C.97) yields [Li , rj ] = i ijk rk ; therefore every vector
operator V satis¬es
[Li , Vj ] = i ijk Vk . (C.98)
The in¬nitesimal rotation formula for an operator which is a vector ¬eld, V = V (r),
contains additional terms due to the argument r:

[Li , Vj (r)] = i {(r — ∇)i Vj (r) + ijk Vk (r)} . (C.99)

Now let us suppose that L is an operator satisfying eqn (C.98) for any choice of V;
then choosing V = L yields eqn (C.94). Therefore any operator L satisfying eqn (C.98)
for all V is the generator of spatial rotations.
In quantum mechanics, there is another kind of angular momentum, called spin,
which has no classical analogue. Particles (or other systems) with spin are described
by n-tuples of wave functions (ψ1 (r) , . . . , ψn (r)). The basic example is the spin-1/2
particle discussed in Appendix C.1.1-A. In the general case, the Hilbert space is a
tensor product, H = Horbital — Hspin , where the orbital (spatial) and spin degrees of
freedom are represented by Horbital and Hspin respectively. Thus the spatial and spin
degrees of freedom are kinematically independent.
¿
Minimal coupling

Since L acts only on the spatial arguments of the wave functions, i.e. on Horbital ,
it can be expressed in the form L = L — Ispin . The spin angular momentum,
S = Iorbital — S acts only on the internal degrees of freedom, and satis¬es the standard
commutation relations
[Si , Sj ] = i ijk Sk . (C.100)
Since L and S act on di¬erent parts of the product space H they must commute:

[Li , Sj ] = [Li — Ispin , Iorbital — Sj ] = [Li , Iorbital ] — [Ispin , Sj ] = 0 , (C.101)

and the total angular momentum J = L + S satis¬es

[Ji , Jj ] = i ijk Jk . (C.102)

This shows that J is the generator of both spatial and spin rotations. In particular,
vector operators will satisfy
[Ji , Vj ] = i ijk Vk . (C.103)
The decomposition of the total angular momentum into the sum of orbital and spin
parts is only possible when L and S commute, i.e. when the spatial and spin degrees
of freedom are kinematically independent.

C.6 Minimal coupling
In minimal coupling, the standard momentum operator ’i ∇ is replaced by

’i ∇ ’ ’i ∇ ’ qA , (C.104)

where A is the vector potential for an external, classical ¬eld. This notion is usually
presented as the simplest way to guarantee the gauge invariance of the quantum theory
for a charge interacting with an external electromagnetic ¬eld; but there is a simpler
explanation, which only involves classical electrodynamics and the correspondence
principle (Cohen-Tannoudji et al., 1977b, Appendix III.3).
The classical Lagrangian for a point particle with charge q interacting with the
classical ¬eld determined by the scalar potential ¦ and the vector potential A is
m2
r ’ q¦ + q r · A .
™ ™
L= (C.105)
2
The canonical momentum p conjugate to r is de¬ned by
‚L
p= = m™ + qA ,
r (C.106)

‚r
so that the kinetic momentum m™ is
r

m™ = p ’ qA .
r (C.107)

The Hamiltonian is de¬ned as a function of r and p by

H (r, p) = p · r ’ L ,
™ (C.108)
Quantum theory


where eqn (C.107) is used to eliminate r in favor of r and p. This leads to

1 2
(p ’ qA) + q¦ .
H= (C.109)
2m
The transition to quantum theory is now made by the correspondence-principle
replacement, p ’ p = ’i ∇. For transverse ¬elds (∇ · A = 0), the quantum Hamil-
tonian is
1 2
(p ’ qA (r)) + q¦
H=
2m
q 2 A (r)2
p2 q
’ A (r) · p +
= + q¦ . (C.110)
2m m 2m
2
For many applications the external ¬eld is weak, so the A (r) -term can be neglected
and the Hamiltonian becomes
p2 q
+ q¦ ’ A (r) · p .
H= (C.111)
2m m
In accord with the classical terminology,

p = ’i ∇ (C.112)

is called the canonical momentum operator, and

pkin = p ’ qA (r) (C.113)

is called the kinetic momentum operator. The velocity operator is v = dr/dt,
and the Heisenberg equation of motion for r (i dr/dt = [r, H]) yields

mv = pkin = p ’ qA (r) . (C.114)

Thus the kinetic momentum operator pkin approaches mvclass in the classical limit,
but the canonical momentum operator p is the generator of spatial translations.
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Index

waist, 80, 220, 227
absorption
coe¬cient, 20 Beer™s law, 20
Bell
of light, 15
additive noise, 432 expectation values, 588
inequality, 578, 590
adiabatic elimination, 377, 427
pair, 637
adjoint
matrix, 650 state measurement, 621
operator, 648 states, 410
alternating tensor, 70, 645 theorem, 589
bipartite system, 201
ampli¬er
birefringence, 399, 676
noise, 505
bit-¬‚ip, 627
noise number, 524
blackbody
noise temperature, 525
cavity, 5
uncertainty principle, 527
radiation, 5
amplitude squeezing/quadrature, 480
bleaching, 460
ancilla, 607
Bloch equation, 375
angular momentum
Bogoliubov transformation, 188, 477
electromagnetic, 100
Bohm singlet state, 580
orbital, 692
bolometers, 6, 266
spin, 692
Boltzmann™s principle, 17
total, 77, 693
annihilation operator, 43, 74 Born
approximation, 31
antilinear, 119
interpretation, 39, 52, 684
antinormal ordering, 167, 179
antire¬‚ection coating, 238, 246 Bose commutation relations, 46
Bose“Einstein statistics, 207
antiresonant Hamiltonian, 352
bosons, 46, 207
antiunitary, 119
bounded operator, 54, 648
atomic coherence, 375
atomic transition operator, 355, 442 box quantization, 81
bra vector, 680
avalanche
Bragg crystal spectrometer, 11
breakdown, 282
broadband detection, 272
multiplication
noise-free, 284
Cn , 201, 646, 681
avoided crossing, 384
c-number, 680
axial vector, 668
Campbell“Baker“Hausdor¬ theorem, 689
balanced canonical
commutation relation, 39“41, 690
beam splitter, 248
momentum, 112, 693
homodyne detector, 300
bare states, 383 momentum operator, 97, 694
Bargmann state, 546 quantization, 69
basis canonically conjugate variables
circular, 685 classical, 39
vector space, 646 quantum, 85, 121, 690
beam carrier
frequency, 88, 219
cleanup, 260
wavevector, 219
in geometric optics, 218
splitter, 247 cascade emission, 24
Casimir e¬ect, 62
balanced, 248
symmetrical, 247 Cauchy“Schwarz inequality, 646
¼
Index

constitutive equations
cavity
linear, 88, 672
frequency, 382
nonlinear, 394, 678
general, 37
continuous spectrum, 648
ideal, 32
controlled
lossy, 428
NOT gate, 633
mode cleaning, 534
modes, 32, 382 sign (or phase) gate, 633
planar, 63, 138, 669 convergent sequence of vectors, 647
convex linear combination, 53, 192, 204
rectangular, 33
convolution theorem, 652
centrosymmetric medium, 393
channeltrons, 282, 388 correlation matrix, 452
correspondence principle, 30, 148, 150
chaotic state, 177
Coulomb gauge, 661
characteristic function, 172
creation operator, 43, 74
antinormally-ordered, 191
cross-Kerr medium, 418, 634
normally-ordered, 183
cross-phase modulation, 418
charge density
cryptography
classical, 670
public key, 616
quantum, 118
circular polarization, 666 quantum, 617
right (left), 55, 667 current density
classical, 670
classical
quantum, 117
bit (cbit), 619
cut-o¬ function, 354
electromagnetic theory, 32
cyclic invariance of the trace, 683
feedforward, 638
nonlinear optics, 678
oscillator, 149 debyes, 134
states, 182 decay rate, 376
click (of a detector), 29 degenerate eigenvalue, 648
degree of degeneracy, 52, 648
closed system, 420
cluster state, 638 degree of freedom
coarse-grained cavity radiation, 7
delta function, 224, 233 mechanical, 40
operator density, 512 degree of polarization, 57
polarization, 90, 678 delta correlated, 434
coherence matrix, 56 delta function, 657
coherence time, 350 density
coherent state of states, 137
operator, 50, 270
diagonal representation, 166
of a single mode, 151, 176 dephasing rate, 376
of a wave packet, 168 detection
coincidence amplitude
basis, 636 one-photon, 213
counting, 286 two-photon, 214
detection, 12 loophole, 597
rate, 25 operator (Mandel), 108
collapse of a cavity state, 386 dichroic mirror, 535
collinear phase matching, 400 dielectrics
complete isotropic and anisotropic, 674, 675
set of commuting observables (CSCO), di¬usion term, 549
102, 684 dipole
set of vectors, 163, 196, 647 approximation, 131
completeness relation, 44, 165, 355, 665, 683 matrix element, 136
compound probability rule, 660 selection rules, 133
Compton discrete quantum trajectory, 573
scattering, 10 dispersion
shift, 14 cancellation e¬ect, 326
wavelength, 14 relation, 674
computational basis, 607 displaced squeezed states, 479
conditional probability, 586, 659 displacement
constant of the motion, 77, 403 ¬eld, 670
½¼ Index

golden rule, 128, 432
operator, 162
rule, 169 statistics, 207
distinct paraxial beams, 221 fermions, 207
distribution, see generalized function ferrite pill, 257
Doppler shift, 134 Feynman
down-conversion, 400 diagrams, 136
dressed paths/processes, 308
photon, 96 rules of interference, 307
states, 384 ¬delity, 612
drift term, 451, 549 ¬eld correlation functions, 123
dynamics, 13 ¬lter function, 274
dynodes, 282 ¬ltered signal, 273
¬nite-dimensional space, 646
¬rst-order perturbation theory, 127
eigenoperators, 355
¬‚uctuation operator, 203, 501, 690
eigenpolarization, 56, 393, 676
eigenspace, 52, 648 ¬‚uorescence spectrum, 462
Fock space, 44, 46
eigenvalue, 648
Fokker“Planck equation, 546
eigenvector, 648
four-port device, see two-channel device
Einstein
A and B coe¬cients, 17, 136 four-wave mixing, 392
Fourier
relation, 453
rule, vii, 98, 258 integral transform, 651
summation convention, 36, 645 series transform, 653
slice theorem, 531
Einstein“Podolsky“Rosen (EPR)
Franson interferometer, 328
paradox, 579
frequency

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