(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