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