Chow T.L. Mathematical Methods for Physicists: A Concise Introduction

In general

A

~

B

~

6

B

~

A

~

. The diÿerence

A

~

B

~

ÿ

B

~

A

~

is called the commutator of

A

~

and

B

~

and is denoted by the symbol 

A

~

;

B

~

]:



A

~

;

B

~



A

~

B

~

ÿ

B

~

A

~

: 5:18

An operator whose commutator vanishe s is called a commuting operator.

The operator equation

B

~

 

A

~



A

~



is equivalent to the vector equation

B

~

ji

 

A

~

ji

for any ji:

And the vector equation

A

~

ji

 

ji

is equivalent to the operator equation

A

~

 

E

~

where

E

~

is the identity (or unit) operator:

E

~

ji



ji

for any ji:

It is obvious that the equation

A

~

  is meaningless.

Example 5.11

To illustrate the non-commuting nature of operators, let

A

~

 x;

B

~

 d=dx. Then

A

~

B

~

f xx

d

dx

f x;

and

B

~

A

~

f x

d

dx

xf x

dx



f  x

df

dx

 f  x

df

dx



E

~



A

~

B

~

f :

Thus,



A

~

B

~

ÿ

B

~

A

~

f xÿ

E

~

f x

or

x;

d

dx



 x

d

dx

ÿ

d

dx

x ÿ

E

~

:

Having de®ned the product of two operators, we can also de®ne an operator

raised to a certain power. For example

A

~

m

ji



A

~

A

~



A

~

ji

:

216

LINEAR VECTOR SPACES

|{z}

m factor

By combining the operations of addition and multiplication, functi ons of opera-

tors can be formed. We can also de®ne functions of operators by their power

series expansions. For example, e

A

~

formally means

e

A

~

 1 

A

~



1

2!

A

~

2



1

3!

A

~

3

:

A function of a linear operator is a linear operator.

Given an operator

A

~

that acts on vector ji, we can de®ne the action of the same

operator on vector hj. We shall use the convention of operating on hj from the

right. Then the action of

A

~

on a vector hjis de®ned by requiring that for any jui

and hvj, we have

f u

hj

A

~

g v

ji

 u

hjf

A

~

v

jig

 u

hj

A

~

v

ji

:

We may write jvijvi and the corresponding bra as hvj. However, it is

important to note that hvjA*hv j.

Eigenvalues and eigenvectors of an operator

The result of operating on a vector with an operator

A

~

is, in general, a diÿerent

vector. But there may be some vector jvi with the property that operating with

A

~

on it yields the same vector jvi multiplied by a scalar, say :

A

~

jvijvi:

This is called the eigenvalue equation for the operator

A

~

, and the vector jvi is

called an eigenvector of

A

~

belonging to the eigenvalue . A linear operator has, in

general, several eigenva lues and eigenvectors, which can be distinguished by a

subscript

A

~

jv

k

i

k

jv

k

i:

The set f

k

g of all the eigenvalues taken together constitutes the spectrum of the

operator. The eigenvalues may be discrete, continuous, or partly discrete and

partly continuous. In general, an eigenvector belongs to only one eigenvalue. If

several linearly independent eigenvectors belon g to the same eigenvalue, the eigen-

value is said to be degenerate, and the degree of degeneracy is given by the number

of linearly independent eigenvectors.

Some special operators

Certain operators with rather special properties play very important roles in

physics. We now consider some of them below.

217

EIGENVALUES AND EIGENVECTORS OF AN OPERATOR

The inverse of an operator

The operator

X

~

satisfying

X

~

A

~



E

~

is called the left inverse of

A

~

and we denote it

by

A

~

ÿ1

L

. Thus,

A

~

ÿ1

L

A

~



E

~

. Similarly, the right inverse of

A

~

is de®ned by the

equation

A

~

A

~

ÿ1

R



E

~

:

In general,

A

~

ÿ1

L

or

A

~

ÿ1

R

, or both, may not be unique and even may not exist at all.

However, if both

A

~

ÿ1

L

and

A

~

ÿ1

R

exist, then they are unique and equal to each

other:

A

~

ÿ1

L



A

~

ÿ1

R



A

~

ÿ1

;

and

A

~

A

~

ÿ1



A

~

ÿ1

A

~



E

~

:

5:19

A

~

ÿ1

is called the operator inverse to

A

~

. Obviously, an operator is the inverse of

another if the corresponding matrices are.

An operator for which an inverse exists is said to be non-singular, whereas one

for which no inverse exists is singular. A necessary and sucient condition for an

operator

A

~

to be non-singular is that corresponding to each vector jui , there

should be a unique vector jvi such that jui

A

~

jvi:

The inverse of a linear operator is a linear operator. The proof is simple: let

ju

1

i

A

~

jv

1

i; ju

2

i

A

~

jv

2

i:

Then

jv

1

i

A

~

ÿ1

ju

1

i; jv

2

i

A

~

ÿ1

ju

2

i

so that

c

1

jv

1

ic

1

A

~

ÿ1

ju

1

i; c

2

jv

2

ic

2

A

~

ÿ1

ju

2

i:

Thus,

A

~

ÿ1

c

1

ju

1

ic

2

ju

2

i 

A

~

ÿ1

c

1

A

~

jv

1

ic

2

A

~

jv

2

i



A

~

ÿ1

A

~

c

1

jv

1

ic

2

jv

2

i

 c

1

jv

1

ic

2

jv

2

i

or

A

~

ÿ1

c

1

ju

1

ic

2

ju

2

i  c

1

A

~

ÿ1

ju

1

ic

2

A

~

ÿ1

ju

2

i:

218

LINEAR VECTOR SPACES

The inverse of a product of operators is the product of the inverse in the reverse

order



A

~

B

~



ÿ1



B

~

ÿ1

A

~

ÿ1

:

5:20

The proof is straightforward: we have

A

~

B

~



A

~

B

~



ÿ1



E

~

:

Multiplying successively from the left by

A

~

ÿ1

and

B

~

ÿ1

, we obtain



A

~

B

~



ÿ1



B

~

ÿ1

A

~

ÿ1

;

which is identical to Eq. (5.20).

The adjoint operators

Assuming that V is an inner-product space, then the operator

X

~

satisfying the

relation

u

hj

X

~

v

ji

 v

hj

A

~

u

ji

* for any jui; jvi2V

is called the adjoint operator of

A

~

and is denoted by

A

~



. Thus

u

hj

A

~



v

ji

 v

hj

A

~

u

ji

* for any jui ; jv i2V: 5:21

We ®rst note that hj

A

~



is a dual vector of

A

~

ji. Next, it is obvious that



A

~









A

~

:

5:22:

To see this, let

A

~





B

~

, then (

A

~







becomes

B

~



, and from Eq. (5.21) we ®nd

v

hj

B

~



u

ji

 u

hj

B

~

v

ji

*

; for any jui; jvi2V:

But

u

hj

B

~

v

ji

*  u

hj

A

~



v

ji

*  v

hj

A

~

u

ji

:

Thus

v

hj

B

~



u

ji

 u

hj

B

~

v

ji

*  v

hj

A

~

u

ji

from which we ®nd



A

~









A

~

:

219

SOME SPECIAL OPERATORS

It is also easy to show that



A

~

B

~







B

~



A

~



:

5:23

For any jui; jvi, hvj

B

~



and

B

~

jvi is a pair of dual vectors; huj

A

~



and

A

~

jui is also a

pair of dual vectors. Thus we have

vhj

B

~



A

~



ujifvhj

B

~



gf

A

~



ujigfuhj

A

~

gf

B

~

vj ig*

 uhj

A

~

B

~

vji*  vhj

A

~

B

~





uji

and therefore



A

~

B

~







B

~



A

~



:

Hermitian operators

An operator

H

~

that is equal to its adjoint, that is, that obeys the relation

H

~



H

~



5:24

is called Hermitian or self-adjoint. And

H

~

is anti-Hermitian if

H

~

ÿ

H

~



:

Hermitian operators have the following important properties:

(1) The eigenvalues are real: Let

H

~

be the Hermitian operator and let jvi be an

eigenvector belonging to the eigenvalue :

H

~

jvijvi:

By de®nition, we have

v

hj

A

~

v

ji

 v

hj

A

~

v

ji

*

;

that is,

* ÿ  vv

jih

 0:

Since hvjvi60, we have

*  :

(2) Eigenvectors belonging to diÿerent eigenvalues are orthogonal: Let jui and

jvi be eigenvectors of

H

~

belonging to the eigenvalues  and ÿ respectively:

H

~

juijui;

H

~

jviÿ jvi:

220

LINEAR VECTOR SPACES

Then

u

hj

H

~

v

ji

 v

hj

H

~

u

ji

*

:

That is,

 ÿ ÿhvjui0 since *  :

But  6 ÿ, so that

hvjui0:

(3) The set of all eigenvectors of a Hermitian operator forms a complete set:

The eigenvectors are orthogonal, and since we can normalize them, this

means that the eigenvectors form an orthonormal set and serve as a basis

for the vector space.

Unitary operators

A linear operator

U

~

is unitary if it preserves the Hermitian character of an

operator under a similarity transformation:



U

~

A

~

U

~

ÿ1







U

~

A

~

U

~

ÿ1

;

where

A

~





A

~

:

But, according to Eq. (5.23)



U

~

A

~

U

~

ÿ1







U

~

ÿ1





A

~

U

~



;

thus, we have



U

~

ÿ1





A

~

U

~





U

~

A

~

U

~

ÿ1

:

Multiplying from the left by

U

~



and from the right by

U

~

, we obtain

U

~





U

~

ÿ1





A

~

U

~



U

~



U

~



U

~

A

~

;

this reduces to

A

~



U

~



U

~



U

~



U

~



A

~

;

since

U

~





U

~

ÿ1







U

~

ÿ1

U

~







E

~

:

Thus

U

~



U

~



E

~

221

SOME SPECIAL OPERATORS

or

U

~





U

~

ÿ1

:

5:25

We often use Eq. (5.25) for the de®nition of the unitary operator.

Unitary operators have the remarkable property that transformation by a uni-

tary operator preser ves the inner product of the vectors. This is easy to see: under

the operation

U

~

, a vector jvi is transformed into the vector jv

0

i

U

~

jvi. Thus, if

two vectors jvi and jui are transformed by the same unitary operator

U

~

, then

u

0

ÿ

þ

v

0



h

U

~

uj

U

~

vi u

hj

U

~



U

~

v

i

 uv

jih

;

that is, the inner product is preserved. In particular, it leaves the norm of a vector

unchanged. Thus, a unitary transformation in a linear vector space is analogous

to a rotation in the physical space (which also preserves the lengths of vectors and

the inner products).

Corresponding to every unitary operator

U

~

, we can de®ne a Hermitian opera-

tor

H

~

and vice versa by

U

~

 e

i"

H

~

;

5:26

where " is a parameter. Obviously

U

~



 e

i"

H

~





=e

ÿi"

H

~



U

~

ÿ1

:

A unitary operator possesses the following properties:

(1) The eigenvalues are unimodular; that is, if

U

~

jvijvi, then jj1.

(2) Eigenvectors belonging to diÿerent eigenvalues are orthogonal.

(3) The product of unitary operators is unitary.

The projection operators

A symbol of the type of juihvj is quite useful: it has all the properties of a linear

operator, multiplied from the right by a ket ji, it gives jui whose magnitude is

hvji; and multiplied from the left by a bra hjit gives hvj whose magnitude is hjui.

The linearity of j uihvj results from the linear properties of the inner product. We

also have

f u

ji

v

hjg



 v

ji

u

hj

:

The operator

P

~

j

 j

ji

j

hj

is a very particular example of projection operator. To

see its eÿect on an arbitrary vector jui, let us expand jui:

222

LINEAR VECTOR SPACES

uji

X

n

j1

u

j

jji; u

j

 jhjui: 5:27

We may write the above as

u

ji



X

n

j1

j

ji

j

hj

ý!

u

ji

;

which is true for all jui. Thus the object in the brackets must be identi®ed with the

identity operator:

I

~



X

n

j1

j

ji

j

hj



X

n

j1

P

~

j

: 5:28

Now we will see that the eÿect of this particular projection operator on jui is to

produce a new vector whose direction is along the basis vector jji and whose

magnitude is hjjui:

P

~

j

u

ji

 j

ji

j

hj

u

i

 j

ji

u

j

:

We see that whatever jui is,

P

~

j

jui is a multiple of jji with a coecient u

j

which is

the component of jui along jji. Eq. (5.28) says that the sum of the projections of a

vector along all the n directions equals the vector itself.

When

P

~

j

 j

ji

j

hj

acts on jji, it reproduces that vector. On the other hand, since

the other basis vectors are orthogonal to jji, a projection operation on any one of

them gives zero (null vector ). The basis vectors are therefore eigenvectors of

P

~

k

with the property

P

~

k

j

ji

 

kj

j

ji

; j; k  1; ...; n:

In this orthonormal basis the projection operators have the matrix form

P

~

1



100

000

.

0

B

@

1

C

A

;

P

~

2



000

010

000

.

0

B

@

1

C

A

;

P

~

N



000 

.

1

0

B

@

1

C

A

:

Projection operators can also act on bras in the same way:

u

hj

P

~

j

 u

hj

j

i

j

hj

 u

j

* j

hj

:

223

SOME SPECIAL OPERATORS

Change of basis

The choice of base vector s (basis) is largely arbitrary and diÿerent representations

are physically equally acceptable. How do we change from one orthonormal set of

base vectors j'

1

i; j'

2

i; ...; j"

n

i to another such set j

1

i; j

2

i; ...; j

n

i? In other

words, how do we generate the orthonomal set j

1

i; j

2

i; ...; j

n

i from the old

set j'

1

i; j'

2

i; ...; j'

n

? This task can be accomplished by a unitary transformation:

j

i

i

U

~

j'

i

ii  1; 2; ...; n: 5:29

Then given a vector X

ji



P

n

i1

a

i

'

i

ji

, it will be transformed into jX

0

i:

jX

0

i

U

~

jXi

U

~

X

n

i1

a

i

'

i

ji



X

n

i1

U

~

a

i

'

i

ji



X

n

i1

a

i



i

ji

:

We can see that the operator

U

~

possesses an inverse

U

~

ÿ1

which is de®ned by the

equation

j'

i

i

U

~

ÿ1

j

i

ii  1; 2; ...; n:

The operator

U

~

is unitary; for, if X

ji



P

n

i1

a

i

'

i

ji

and Y

ji



P

n

i1

b

i

'

i

ji

, then

X

j

Y

hi



X

n

i;j1

a

i

*

b

j

'

i

j'

j

ÿ



X

n

i1

a

i

*

b

i

; hUX

j

UY

i



X

n

i;j1

a

i

*

b

j



i

j

j

ÿ



X

n

i1

a

i

*

b

i

:

Hence

U

ÿ1

 U



:

The inner product of two vectors is independent of the choice of basis which

spans the vector space, since unitary transformations leave all inner products

invariant. In quantum mechanics inner products give physically observable quan-

tities, such as expe ctation values, probabilities, etc.

It is also clear that the matrix representation of an operator is diÿerent in a

diÿerent basis. To ®nd the eÿect of a change of basis on the matrix representation

of an operator, let us consider the transformation of the vector jXi into jYi by the

operator

A

~

:

jYi

A

~

jXi: 5:30

Referred to the basis j'

1

i; j'

2

i; ...; j'i; jXi and jY i are given by

jXi

P

n

i1

a

i

'

i

j

and jYi

P

n

i1

b

i

j'

i

i, and the equation jYi

A

~

jXi becomes

X

n

i1

b

i

'

i

ji



A

~

X

n

j1

a

j

'

j

þ



:

224

LINEAR VECTOR SPACES

Multiplying both sides from the left by the bra vector h'

i

j we ®nd

b

i



X

n

j1

a

j

'

i

hj

A

~

'

j

þ





X

n

j1

a

j

A

ij

: 5:31

Referred to the basis j

1

i; j

2

i; ...; j

n

i the same vectors jXi and jYi are

jXi

P

n

i1

a

0

i



i

j

, and jYi

P

n

i1

b

0

i

j

i

i, and Eqs. (5.31) are replaced by

b

0

i



X

n

j1

a

0

j



i

hj

A

~



j

þ





X

n

j1

a

0

j

A

0

ij

;

where A

0

ij

h

i

j

A

~



j

i

þ

, which is related to A

ij

by the following relation:

A

0

ij

 

i

hj

A

~



j

þ



 U'

i

hj

A

~

U'

j

þ



 '

i

hj

U*

A

~

U '

j

þ



U*

A

~

U

ij

or using the rule for matrix multiplication

A

0

ij

 

i

hj

A

~



j

þ



U*

A

~

U

ij



X

n

r1

X

n

s1

U

ir

*A

rs

U

sj

: 5:32

From Eqs. (5.32) we can ®nd the matrix representation of an operator with

respect to a new basis.

If the operator

A

~

transforms vector jXi into vector jYi which is vector jXi itself

multiplied by a scalar : jYijXi, then Eq. (5.30) becomes an eigenvalue

equation:

A

~

jXi jXi:

Commuting operators

In general, operators do not commute. But commuting operators do exist and

they are of importance in quantum mechanics. As Hermitian ope rators play a

dominant role in quantum mechanics, and the eigenvalues and the eigenvectors of

a Hermitian operator are real and form a complete set, respectively, we shall

concentrate on Hermitian operators. It is straightforward to prove that

Two commuting Hermitian operators possess a complete ortho-

normal set of common eigenvectors, and vice versa.

If

A

~

and

A

~

jvijvi are two commuting Hermitian operators, and if

A

~

jvijvi; 5:33

then we have to show that

B

~

jviÿjvi: 5:34

225

COMMUTING OPERATORS

Chow T.L. Mathematical Methods for Physicists: A Concise Introduction

Подождите немного. Документ загружается.