Appel W. Mathematics for Physics and Physicists

Подождите немного. Документ загружается.

Hermitian operators; self-adjoint operators 397

Remark 14.66 Concerning point iv), there is nothing a priori that ensures that such a basis of

eigenvectors exists.

PROPOSITION 14.67 The residual spectrum of a self-adjoint operator A is empty.

Proof. Let λ ∈ σ

(A). Then λ ∈ σ

†

) = σ

(A) since A = A

†

; but the eigenvalues

of A are real, hence λ = λ and therefore λ ∈ σ

(A), which contradicts the deﬁnition

of the residual spectrum.

14.4.c Generalized eigenvectors

If A is an operator and λ is a generalized eigenvalue, we are going to try to

solve the equation AT = λT in the sense of (tempered) distr ibutions. The

ﬁrst thing to do is to deﬁne AT . We have already seen the method: start by

looking at what happens in the case of a regular distribution T

associated

with a function ψ ∈ S , that is, deﬁned by

, ϕ

= 〈ψ|ϕ〉 =

+∞

−∞

ψ(x) ϕ(x) dx.

If A is a symmetric operator, we have

∀ϕ ∈ S 〈Aψ



ϕ〉 = 〈ψ|Aϕ〉,

hence ∀ϕ ∈ S 〈AT



ϕ〉 =



Aϕ

This property can be extended, as long as Aϕ is deﬁned for all ϕ ∈ S and is

also in S :

DEFINITION 14.68 (Image of a distribution by a symmetric operator) Let A

be a symmetric operator, such that

• S ⊂ D

;

• S is stable under A;

• A is continuous on S (for its topology).

Then A is of class SS

SS . For any tempered distribution T ∈ S

′

, the tempered

distribution AT is deﬁned by

∀ϕ ∈ S 〈AT |ϕ〉 = 〈T |Aϕ〉.

 Exercise 14.1 Prove that AT is indeed a tempered distribution.

DEFINITION 14.69 (Generalized eigenvector) Let A be a symmetric operator

of class S . A nonzero tempered distribution T is an eigendistribution or a

generalized eigenvector, associated to the eigenvalue λ, if and only if

AT = λT ,

in other words if 〈T |Aϕ〉 = λ 〈T |ϕ〉 for any ϕ ∈ S .

398 Bras, kets, and all that sort of thing

The main result is the following:

THEOREM 14.70 (Spectral theorem) For any self-adjoint operator of a Hilbert

space H which is of class S , there exists an orthonormal basis of eigendistributions

which are associated to real eigenvalues or generalized eigenvalues.

Remark 14.71 There exists a much more general version of this theorem, wh ich applies to any

self-adjoint operator. This generalized spectral theorem states that there always exists a spectral

family, but this notion is beyond the scope of this book. It is discussed in many books, for

instance [71, §III.5, 6], [95, p. 424], and [38, vol. 4, ch.1, §4].

PROPOSITION 14.72 The momentum operator P on the space L

(R) is of class S .

The family



〈p| ; p ∈ R



is a generalized orthonormal basis of generalized eigenvec-

tors for P .

Proof. We already know, by Theorem 14.25, that this family is a generalized basis.

The orthonormality is the statement that

∀p, p

′

∈ R 〈p| p

′

〉 = δ(p − p

′

We now check that, denoting by Ω

the regular distribution associated to the function

(x) = (2π }h)

1/2

iπx/ }h

, we have P Ω

= pΩ

Indeed, let ϕ ∈ S . Then we have



P Ω







Ω



P ϕ



by Deﬁnition 14.68



Ω



−i }h ϕ

′



= −i }h

′

(p)

= p eϕ(p) since F [ϕ

′

] (p) = (i p/ }h) eϕ(p)

= p



Ω







p Ω





since p is real.

Remark 14.73 The distribution Ω

is, up to normalization, the limit of the ω

(ǫ)

deﬁned in

Example 14.52.

PROPOSITION 14.74 The operator X on L

(R) is of class S . Moreover, the

family



〈x| ; x ∈ R



is a genera lized orthonormal basis of generalized eigenvectors

for X .

Proof. It is only necessary to c heck that X δ(x − x

) = x

δ(x − x

) in the sense of

Deﬁnition 14.68. For any ϕ ∈ S , we have indeed



X δ(x − x

)







δ(x − x

)



X ϕ



= (X ϕ)(x

) = x

ϕ(x

) = x



δ(x − x

)





14.4.d “Matrix” representation

Let A b e an operator on H, with dense domain as usual. Let (ϕ

)

n∈N

be an

orthonormal basis of H, all elements of which are also in D

. The associated

matrix elements a re deﬁned by

i,k

def

= 〈ϕ

|A ϕ

〉.

The question is: is it true or not that the knowledge of the inﬁnite “matrix ”

M = (a

)

i,k

is sufﬁcient to reconstruct entirely the operator A?

Hermitian operators; self-adjoint operators 399

DEFINITION 14.75 The operator A has a matrix representation as sociated

with (ϕ

)

if and only if

∀ψ =

∞

k=0

Aψ =

∞

i=0



∞

k=0



. (14.1)

Assume that A has a matrix representation. Using Dirac’s notation, we

have x

= 〈ϕ

|ψ〉; still following Dirac, deﬁne

〈ϕ

|A |ϕ

〉

def

= 〈ϕ

|Aϕ

〉

and derive

A |ψ〉 =

∞

i=0

∞

k=0

|ϕ

〉〈ϕ

|A |ϕ

〉〈ϕ

|ψ〉,

which is again the closure relation. Symbolically, this is written:

A =

i,k

|ϕ

〉〈ϕ

The difﬁculty is that these formulas are not always correct, s ince the r ight-

hand side of (14.1) is not always correctly deﬁned for all ψ ∈ D

. It is easy

to see that, if A is bounded, then by continuity everything works out. More

generally:

THEOREM 14.76 (Matrix representation) I f A be a self-adjoint operator, then it

has a matrix representation associated to any orthonormal basis of D

This property is also true if A is bounded, or if it is hermitian and c losed.

As an example, th e operators of orthogonal projection on a closed subspace

are self-adjoint operators which admit a matrix representation:

THEOREM 14.77 Let (ω

)

k∈N

be an orthonormal family, and let M be the closure

of Vect{ω

; k ∈ N}. Then M is a closed subspace. Let M

⊥

be its orthogonal

complement. Any vector ψ ∈ H can be decomposed in a unique way as

ψ = ψ

+ χ with ψ

∈ M and χ ∈ M

⊥

This is where the trap lies for those w h o manipulate the notation blindly. The deﬁnition

〈ϕ|A

def

= 〈A

†

ϕ|, which is often found, permits two readings of the e xpression 〈ϕ|A |ψ〉:

• as



†





— which imposes that |ϕ〉 ∈ D

†

and |ψ〉 ∈ H;

• as 〈ϕ|Aψ〉 — which imposes that |ϕ〉 ∈ H and |ψ〉 ∈ D

Those are not quite equivalent — always because of the issue of the exact domain of an op erator.

Note that , for a self-adjoint operator, there is no ambiguity; for a hermitian operator, mistakes

can arise because of this [41].

For instance, i f

∞

i=0

< +∞ for all k ∈ N, the operator A : ϕ

7→

∞

i=0

deﬁnes, by density of Vect{ϕ

; i ∈ N}, a closed operator, which is hermitian if a

= a

400 Bras, kets, and all that sort of thing

The operator Π which associates ψ

to ψ is the orthogonal projector on M. It

satisﬁes Π

= Π and Π

†

= Π. In matrix representation it is given by

Π =

∞

k=0

|ω

〉〈ω

Let A be a self-adjoint operator, and assume that there is an orthonormal

Hilbert basis (ϕ

)

n∈N

of eigenvectors of A, with corresponding eigenvalues

(λ

)

n∈N

(recall th at this is by no means always the case). Then

A =

∞

n=0

|ϕ

〉〈ϕ

that is,

∀ψ ∈ D

A |ϕ〉 =

∞

n=0

〈ϕ

|ψ〉 |ϕ

〉,

and

ψ ∈ D

if and only if



〈ϕ

|ψ〉



converges.

With this th e formal calculus of the operator A can be deﬁned:

DEFINITION 14.78 (Function of an operator) Let A be a self-adjoint operator,

(ϕ

)

n∈N

an orthonormal Hilbert basis of eigenvectors of A, associated to the

eigenvalues (λ

)

n∈N

. Let F : R → R be a measurable function. The operator

F (A) is deﬁned by

F (A) =

∞

n=0

F (λ) |ϕ

〉〈ϕ

with domain

F(A)

ψ ∈ H ;

∞

n=0

F (λ)

|〈ϕ

|ψ〉|

< +∞

All the ϕ

are in D

F(A)

, hence D

F(A)

is also dense. In fact we have [95, §5.8]:

THEOREM 14.79 The operator F (A) is self-adjoint.

Since the identity operator is obviously self-adjoint and deﬁned on H,

for a ny orthonormal Hilbert basis (ϕ

)

n∈N

and any sequence (β

)

n∈N

of real

numbers, the operator

def

∞

n=0

|ϕ

〉〈ϕ

is a self-adjoint operator.

This technique can provide the extension of an operator which is merely

hermitian, but admits an orthonormal basis of eigenvectors, to a self-adjoint

operator — deﬁned therefore on a larger domain, and wh ich is the only one

for which many computat ions make sense.

Hermitian operators; self-adjoint operators 401

For instance, consider the operator H

deﬁned on D

= D(R

) (space of

∞

functions with bounded suppoert), deﬁned by

∀ψ ∈ D H

ψ(x) =



−

△+V (x)



ψ(x),

where V is a given function. In many cases, H

is essentially self-adjoint,

that is, th e closure H

is self-adjoint; this closure of H

is then given by

= H

†

. The self-adjoint extension H = H

, is the hamiltonian operator

of the system. It is this operator which is observable. In practice, it is

obtained by ﬁnding an orthonormal basis (ϕ

)

n∈N

for H

and applying the

method above.

Example 14.80 (Hamiltonian for the harmonic oscillator) Consider a harmonic oscillator with

mass m and pulsation ω. We ﬁrst deﬁne a formal hamiltonian H

on D

= S by

|ψ〉 =

|ψ〉+

mω

|ψ〉.

This operator is hermitian, and has a Hilbert basis (ϕ

)

n∈N

such that

|ϕ

〉 = E

|ϕ

〉 with E



n +



}hω.

The functions ϕ

are of the type exp(−mω

/2 }h) h

(x), where h

is a Hermite polynomial

of degree n [20, Ch. V §C.2]. The operator H

is not self-adjoint.

Let us now deﬁne the operator H by

H =

∞

n=0

|ϕ

〉〈ϕ

|, i.e., H |ψ〉 =

∞

n=0

〈ϕ

|ψ〉|ϕ

〉

for any function |ψ〉 such as the series



〈ϕ

|ψ〉



converges.

The operator H i s self-adjoint; it is an extension of H

, which is called the hamiltonian

of the harmonic oscillator.

An example of this is given in Exercise 14.5.

14.4.e Summary of properties of the operators P and X

PROPOSITION 14.81 (Oper a tor P on L

(RR

RR)) The operator P on L

(R) is self-

adjoint (and hence closed). It is deﬁned on

= {ϕ ∈ H ; ϕ

′

∈ H}.

The discrete spectrum of P is empty; its continuous spectrum is equal to R. The family



〈p| ; p ∈ R



is a generalized basis of eigenv ectors such that





′



= δ(p − p

′

), (orthonormal basis)

Id =

+∞

−∞

|p〉〈p| dp. (closure relation)

Proof. See Example 14.52 for the conti nuous sp ectrum and Theorem 14.72 for the

basis of eigenvectors.

402 Bras, kets, and all that sort of thing

PROPOSITION 14.82 (Oper a tor P on L

[0, a])

The operator P on L

[0, a] is hermitian (symmetric), but not self-adjoint. Its domain



ϕ ∈ H ; ϕ

′

∈ H and ϕ(0) = ϕ(a) = 0



P is closed (and therefore a dmits matrix representations).

The discrete spectrum of P is empty; its continuous spectrum is also empty and its

residual spectrum is equal to C.

Proof. The ﬁrst property has already been proved in Proposition 14.64.

Before trying to prove the absence of any eigenvalue, even generalized, we can try

to reason “physically.” A vector such that P ϕ is close to λϕ is characterized by a

very small “uncertainty” ∆p with respect to the momentum, which, according to

Heisenberg’s uncertainty principle is only possible if ∆x is very large. But, here, ∆x

is bounded by a.

Such an argument is not simply h euristic: it can be justiﬁed by noting that P

and X (on L

(R)) are unitarily equivalent via the Fourier transform, namely, we have

P = F X F

−1

. Let now p

be any real number; we will show that p

is not a generalized

eigenvalue. Let ϕ be a function with norm 1. The Fourier transform

ψ of (P − p

)ϕ =

−i }hϕ

′

− p

ϕ is p

′

7→ (p − p

) eϕ(p). The Parseval-Plancherel identify shows that



(P − p

)ϕ



(p − p

) eϕ(p)



The norm squared



is si mply the average value, in the state corresponding to ϕ,

of the quantity (p −p

)

, which was denoted



(p − p

)



in the section on u ncertainty

relations. It can be checked that the quantity



(p − p

)



is smallest for p

= 〈p〉,

hence



ψ(p)





(p − p

)



¾ (∆p)

By the uncertainty relations (Theorem 13.17), we know that ∆p ¾ }h/2∆x ¾ }h/2a,

which ﬁnally implies



Consequently, it is not possible to make



p 7→ (p − p

) eϕ(p)



(P − p

)ϕ



small as possible, so that p

is not a generalized eigenvalue.

One easily checks that the domain of P

†

is D

†

= {ϕ ∈ H ; ϕ

′

∈ H}, and that any

complex λ is an eigenvalue of P

†

, with associated eigenvector ϕ

: x 7→ e

iλx/ }h

. This

proves that σ

(P ) = C.

PROPOSITION 14.83 (Oper a tor X on L

(RR

RR)) The operator X on L

(R) is self-

adjoint; its domain is

= {ϕ ∈ L

(R) ; x 7→ x ϕ(x) ∈ L

(R)}.

The discrete spectrum of X is empty; its continuous spectrum is R. The family



〈x| ; x ∈ R



is a generalized basis of generalized eigenvectors such that





′



= δ(x − x

′

), (orthonormal basis)

Id =

+∞

−∞

|x〉〈x| dx. (closure relation)

Exercises 403

Proof. See Theorem 14.62 to show that X is self-adjoint. See Remark 14.50 for the

spectrum.

PROPOSITION 14.84 (Oper a tor X on L

[0, a]) The operator X on L

[0, a] is

self-adjoint and bounded. Its domain is D

= L

[0, a] . The discrete spectrum of X

is empty. Its continuous spectrum is [0, a]. The family



〈x| ; x ∈ [0, a]



is a generalized basis of eigenvectors of X on L

[0, a] .

Proof. See Theorem 14.62 to show that X is self-adjoint. The computation of the

spectrum is le ft as an exercise; the proof in Remark 14.50 can be used as a clue.

EXERCISES

 Exercise 14.2 Check that any eigenvector is also an ei gendistribution.

 Exercise 14.3 Let A be a linerar operator deﬁned on the subspace D

= W . For any

complex number λ, the operator A

= A −λ Id is also deﬁned on W . One and only one of

the following four cases applies for λ (see, e.g., [94]):

• A

is not injective; then λ is an eigenvalue, o r equivalently λ belongs to the discrete

spectrum.

• A

is injective, Im(A

) is dense but the inverse

of A

is not continuous; then λ is a

generalized eigenvalue and λ belongs to the continous spectrum.

• A

is injective but Im(A

) is not dense; then λ belongs to the residual spectrum;

• A

is injective, Im(A

) is dense and A

−1

is continuous (and hence may be extended by

continuity to an operator deﬁned on the whole of H); then λ belongs to the resolvant

set of A.

Show th at those deﬁniti ons agree with those in Deﬁnitions 14.46 and 14.49.

Note that in some books (for instance [4]), the deﬁnitions somewhat different, and the

discrete and continuous spectrum, for instance, are not necessarily disjoint. (This may actually

be desirable to account for natural situations, such as some hyperbolic billiards arising in

“arithmetic quantum chaos,” where there is a seemingly chaotic discrete spectrum embedded

in a very regular continuous spectrum; see, e.g., [78].)

 Exercise 14.4 (Continuous spectrum of X) Consider the Hilbert space L

[a, b ]. Show that

X h as no eigenvalue, but that any λ ∈ [a, b ] is a generalized eigenvalue, that is, σ

(X ) = ∅

and σ

(X ) = [a, b].

 Exercise 14.5 (A “paradox” of quantum mechanics, following F. Gieres [41])

Consider a particle, in an inﬁnite potential sink, constrained to remain in the interval [−a, a].

Physical considerations impose that the wave functions satisfy ϕ(−a) = ϕ(a) = 0. The

hamiltonian is the operator on L

[−a, a] deﬁned by H ϕ = −}h

′′

/2m. Assume H i s a

self-adjoint operator.

The target space of A

can be restricted to Im(A

); then A

is a bijection from W

to Im(A

404 Bras, kets, and all that sort of thing

i) Solve the eigenvalue equation H ψ = Eψ, associated to th e boundar y conditions

ψ(±a) = 0; deduce from this the spe ctrum (E

)

n∈N

∗

and an orth onormal basis of

eigenfunctions (ϕ

)

n∈N

∗

ii) Consider the wave function given by ψ(x) = µ(a

− x

), wh ere µ is a normalizing

factor so that 〈ψ|ψ〉 = 1. Show th at 〈H ψ|H ψ〉 > 0. Isn’t it the case however that

ψ = 0 and hence







= 0? How should the average value of the energy

square in the state ψ be computed? Comment.

 Exercise 14.6 (Uncertainty principle) Let A and B be two self-adjoint operators on H (i.e.,

“observables”). Let ψ ∈ H be a normed vector (i.e., a “physi cal state”) such that

ψ ∈ D

∩ D

, Aψ ∈ D

, and Bψ ∈ D

The average value of A and the uncertainty on A in the state ψ are deﬁned by

def

= 〈ψ|Aψ〉 and ∆a = kAψ − aψk.

Show th at

∆a ·∆b ¾







[A, B]ψ





, where [A, B]

def

= A B − B A.

Check that [X , P ] = i }h Id and deduce from this that ∆x ·∆p ¾ }h/2.

SOLUTIONS

 Solution of exercise 14.5.

i) Solving the eigenvalue equation leads to the discrete sp ectrum given by:

8ma

(n ¾ 1)

associated with the eigenfunctions

(x) =

cos

p

2mE



cos



nπ x



if n is odd,

(x) =

sin

p

2mE



sin



nπ x



if n is even.

It is easy to check that the sequence ( ϕ

)

n∈N

∗

is a complete orthonormal system of

eigenfunctions of H .

ii) First compute that µ =

15/4a

5/2

. Then another computation shows that

kH ψk

= 〈H ψ|H ψ〉 =

15 }h

Moreover, since the fourth derivative of ψ is zero, it is tempting to write H

ψ = 0 and,

consequently,







= 0; which is quite annoying since H is self-adjoint, and one

would expect to have identical results.

The point is that o ne must be more careful with the deﬁnition of the operator H

Indeed, the domain of H is

= {ϕ ∈ H ; ϕ

′′

∈ H and ϕ(±a) = 0};

Solutions of exercises 405

hence H ψ /∈ D

and therefore it is not permitted to compute H

ψ as the c o mposition

H (H ψ). (It may be checked that the elements of D

must satisfy ϕ

′′

(±a) = 0.) On

the other hand, using the orthonormal Hi l bert basis (ϕ

)

n∈N

∗

, we can write

H =

∞

n=1

|ϕ

〉〈ϕ

and, using the techniques of Section 14.4.d, D eﬁnition 14.78, it is possible to deﬁne a

new operator, denoted H

, by

def

∞

n=1

|ϕ

〉〈ϕ

Notice that since the eigenfunctions ϕ

are i n the domain of H

, we have

∀n ¾ 1 H

= H

This new operator is thus an extension of the previo u s one. Its domain is the subspace

of functions ψ such that



〈ϕ

|ψ〉



< +∞. Then one checks that the function

ψ, which did not belong to the domain

of H

, is indeed in the domain of H

This extension is self-adjoint. A (boring) computation with Fourier serie s shows that

∀n ¾ 1







def

∞

n=1

〈ψ|ϕ

〉〈ϕ

|ψ〉 =

15 }h

i.e., that







= 〈H ψ|H ψ〉 as it should be.

 Solution of exercise 14.6. First of all, notice that a and b are real numbers si nce A and B

are hermitian operators (Theorem 14.65). Also, notice that





(A B − B A) ψ



Since a and b are real, we can expand



(A −a)ψ



(B − b)ψ



= 〈Aψ|Bψ〉−ab − ba + ab kψk

= 〈Aψ|Bψ〉−ab,

and obtain





(A B − B A) ψ



= 2 Im



(A −a)ψ



(B − b)ψ



The conclusion now follows using the Cauchy-Schwarz inequality:

∆a ·∆b ¾





(A − a)ψ



(B − b)ψ







(A − a)ψ



(B − b)ψ





Im 〈Aψ|Bψ〉







[A, B] ψ





This is of course linked to th e Fourier series of ψ, which converges uniformly but cannot

be differentiated twice termwise.