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

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is a basis for the vector space M

of 2  2 matrices. To see that S spans M

, note

that a typical 2  2 vector (matrix) can be written as

ý!

 a

ý!

 b

ý!

 c

ý!

 d

ý!

 a 1

 b 2

 c 3

 d 4

To see that S is linearly independent, assume that

a 1

 b 2

 c 3

 d 4

 0

;

that is,



 b



 c



 d







;

from which we ®nd a  b  c  d  0 so that S is linearly independent.

We now come to the dimensionality of a vector space. We think of space

around us as three-dimensional. How do we extend the notion of dimension to

a linear vector space? Recall that the three-dimensional Euclidean space E

spanned by the three base vectors: e

1; 0; 0, e

0; 1; 0, e

0; 0; 1.

Similarly, the dimension n of a vector space V is de®ned to be the number n of

linearly independent base vector s that span the vector space V. The vector space

will be denoted by V

R if the ®eld is real and by V

C if the ®eld is complex.

For example, as shown in Example 5.6, 2  2 matrices form a four-dimensional

vector spac e whose base vectors are





; 2





; 3





; 4





;

since any arbitrary 2  2 matrix can be written in terms of these:



 a 1

 b 2

 c 3

 d 4

If the scalars a; b; c; d are real, we have a real four-dimensi onal space, if they are

complex we have a complex four-dimensional space.

Inner product spaces (unitary spaces)

In this section the structure of the vector space will be greatly enriched by the

addition of a numerical function, the inner product (or scalar product). Linear

vector spaces in which an inner product is de®ned are called inner-product spaces

(or unitary spaces). The study of inner-product spaces enables us to make a real

juncture wi th physics.

206

LINEAR VECTOR SPACES

In our earlier discus sion, the inner product of two vectors in E

was de®ned by

Eq. (5.6), a generalization of the inner product of two vectors in E

. In a general

linear vector space, an inner product is de®ned axiomatically analogously with the

inner product on E

. Thus given two vectors U

and W



i1

; W



i1

; 5:9

where U

and W

are express ed in terms of the n base vectors i

, the inner

product, denoted by the symbol UW

jih

, is de®ned to be

hUW



i1

j1

jih

: 5:10

is often called the pre-factor and W

the post-factor. The inner product obeys

the following rules (or axioms):

B.1 UW

 WU

(skew-symmetry);

B.2 UjU

 0;  0 if and only if U

 0

(positive semide®niteness);

B.3 UX

 W

ji

 UX

 UW

jihhh

(additivity);

B.4 aU W

 a* UW

; UbW

 bUW

jihhhh

(homogeneity);

where a and b are scalars and the asterisk (*) denotes complex conjugation. Note

that Axiom B.1 is diÿerent from the one for the inner product on E

: the inner

product on a general linear vector space depends on the order of the two factors

for a complex vector space. In a real vector space E

, the complex conjugation in

Axioms B.1 and B.4 adds nothing and may be ignored. In either case, real or

complex, Axiom B.1 implies that UU

jih

is real, so the inequality in Axiom B.2

makes sense.

The inner product is linear with respect to the post-factor:

aW  bX

 aU

 bU

;

and anti-linear with respect to the prefactor,

aU  bX W

 a* UW

 b* XW

hhh

Two vectors are said to be orthogonal if their inner product vanishes. And we

will refer to the quantity hUU

1=2

kU k as the norm or length of the vector. A

normalized vector, having unit norm, is a unit vector. Any given non-zero vector

may be normalized by dividing it by its length. An orthonormal basis is a set of

basis vectors that are all of unit norm and pair-wise orthogonal. It is very handy

to have an orthonormal set of vectors as a basis for a vector space, so for hijji in

Eq. (5.10) we shall assume

 



1 for i  j

0 for i 6 j

(

;

207

INNER PRODUCT SPACES (UNITARY SPACES)

then Eq. (5.10) redu ces to









ý!



: 5 :11

Note that Axiom B.2 implies that if a vector U

is orthogonal to every vector

of the vector space, then U

 0: since U  0

ijh

for all jibelongs to the vector

space, so we have in particular UUji0h .

We will show shortly that we may construct an orthonormal basis from an

arbitrary basis using a technique known as the Gram±Schmidt orthogonalization

process.

Example 5.7

Let jUi3 ÿ 4ij1i5 ÿ 6ij2i and jWi1 ÿ ij1i2 ÿ 3ij 2 i be two vec-

tors expanded in terms of an orthonormal basis j1i and j2i. Then we have, using

Eq. (5.10):

3  4i3 ÿ 4i5  6i5 ÿ 6i86;

1  i1 ÿ i2  3i2 ÿ 3i 15;

3  4i1 ÿ i5  6i2 ÿ 3i35 ÿ 2i  W

Example 5.8

A and

B are two matrices, where

A 

ý!

;

B 

ý!

;

then the following formula de®nes an inner product on M



 a

 a

To see this, let us ®rst expand

A and

B in terms of the following base vectors

1ji



; 2ji



; 3ji



; 4ji



;

A  a

 a

;

B  b

 b

The result follows easily from the de®ning formula (5.10).

Example 5.9

Consider the vector jUi, in a certain ortho normal basis, with components



1  i



 i

ý!

; i 



ÿ1

208

LINEAR VECTOR SPACES

We now expand it in a new orthonormal basis je

i; je

i with components







; e





ÿ1



To do this, let us write

 u

 u

and determine u

and u

. To determine u

, we take the inner product of both sides

with he

 e





p 11



1  i



 i

ý!





1 



 2i;

likewise,





1 ÿ



:

As a check on the calculation, let us compute the norm squared of the vector and

see if it equals j1  ij

j



 ij

 6. We ®nd

 u



1  3  2



 4  1  3 ÿ 2



6:

The Gram±Schmidt orthogonalization process

We now take up the Gram±Schmidt orthogonalization method for converting a

linearly independent basis into an orthonormal one. The basic idea can be clearly

illustrated in the following steps. Let j1i; j 2 i ; ...; jii; ... be a linearly independent

basis. To get an orthonormal basis out of these, we do the following:

Step 1. Rescale the ®rst vector by its own length, so it becomes a unit vector.

This will be the ®rst basis vector.



jijj

;

where 1

jijj





. Clearly



jijj

 1:

Step 2. To construct the second member of the orthonormal basis, we subtract

from the second vector j2i its projection along the ®rst, leaving behind

only the part perpendicular to the ®rst.

 2

ÿ e

209

THE GRAM±SCHMIDT ORTHOGONALIZATION PROCESS

Clearly

jII

 e

ÿ e

 0; i:e:; IIj?je

Dividing jIIi by its norm (length), we now have the second basis vector

and it is orthogonal to the ®rst base vector je

i and of unit length.

Step 3. To construct the third member of the orthonormal basis, consider

jIIIij3iÿje

ihe

jIIIiÿje

jIIIi

which is orthogonal to both je

i and je

i. Dividing by its norm we get

Continuing in this way, we will obtain an orthonormal basis je

i; je

i; ...; je

The Cauchy±Schwarz inequality

If A and B are non-zero vectors in E

, then the dot product gives A  B  AB cos ,

where  is the angle between the vectors. If we square both sides and use the fact

that cos

  1, we obtain the inequality

A  B

 A

or A  B

 AB:

This is known as the Cauchy±Schwarz inequality. There is an inequality corre-

sponding to the Cauchy±Schwarz inequality in any inner-product space that

obeys Axio ms B.1±B.4, which can be stated as

ijj

jUj W

; U





etc:; 5:13

where jUi and jWi are two non-zero vectors in an inner-product space.

This can be proved as follows. We ®rst note that, for any scalar , the following

inequality holds

0  U  W

U  W

ijj

 U  W

U  W

 U

 W

 U

W

 W

W

 U

* V

  U

 

Now let   hUjWi*=jhUjWij, with  real. This is possible if jWi60, but if

hUjWi0, then Cauchy±Schwarz inequality is trivial. Making this substi tution

in the above, we have

0  Ujj

 2 UhjWijj

Wjj

This is a quadratic expression in the real variable  with real coecients.

Therefore, the discriminant must be less than or equal to zero:

4 U

ijj

ÿ4 U

 0

210

LINEAR VECTOR SPACES

ijj

 U

;

which is the Cauchy±Schwarz inequality.

From the Cauchy±Schwarz inequality follows another important inequality,

known as the triangle inequality,

U  WjjUjjWjj: 5:14

The proof of this is very straightforward. For any pair of vectors, we have

U  W

 U  W

U  W

 U

 W

 U

 W

 U

 W

 2 U

ijj

 U

 W

 2 U

jj

 W



from which it follows that

U  W

 U

 W

If V denotes the vector space of real continuous functions on the interval

a  x  b, and f and g are any real continuous functions, then the following is

an inner product on V:



f xgxdx:

The Cauchy±Schwarz inequal ity now gives

f xgxdx





xdx

or in Dirac notation

ijj

 f

Dual vectors and dual spaces

We begin with a technical point regarding the inner product hujvi. If we set

jvijwiÿjzi;

then

hujvihujwiÿhujzi

is a linear function of  and ÿ. However, if we set

juijwiÿjzi;

211

DUAL VECTORS AND DUAL SPACES

then

hujvihvjui*  *hvjwi*  ÿ*hvjzi*  *hwjviÿ*h zjvi

is no longer a linear function of  and ÿ. To remove this asymmetry, we can

introduce, besides the ket vectors ji, bra vectors hj which form a diÿerent vector

space. We will assume that there is a one-to-one correspondence between ket

vectors ji, and bra vectors hj. Thus there are two vector spaces, the space of

kets and a dual space of bras. A pair of vectors in which each is in correspondence

with the other will be called a pair of dual vectors. Thus, for example, hvj is the

dual vector of jvi. Note they always carry the same identi®cation label.

We now de®ne the multiplication of ket vectors by bra vectors by requiring

hujjvihujvi:

Setting

 w

*  z

ÿ*;

we have

hujvi*hwjviÿ*hzjvi;

the same result we obtained above, and we see that w

*  z

ÿ* is the dual

vector of jwiÿjzi.

From the above discussion, it is obvious that inner products are really de®ned

only between bras and kets and hence from elements of two distinct but related

vector spaces. There is a basis of vectors jii for expanding kets and a similar basis

hij for expanding bras. The basis ket jti is represented in the basis we are using by

a column vector with all zeros except for a 1 in the ith row, while the basis hij is a

row vector with all zeros except for a 1 in the ith column.

Linear operators

A useful concept in the study of linear vector spaces is that of a linear transfor-

mation, from which the concept of a linear operator emerges naturally. It is

instructive ®rst to review the concept of transformation or mapping. Given vector

spaces V and W and function

,if

associates each vector in V with a unique

vector in W, we say

maps V into W, and write

: V ! W.If

associates the

vector jwi in W with the vector jvi in V, we say that jwi is the image of jvi under

and write jwi

jvi. Further,

is a linear transformation if:

(a)

juijvi 

jui

jvi for all vectors jui and jvi in V.

(b)

kjvi  k

jvi for all vectors jvi in V and all scalars k.

We can illustrate this with a very simple example. If jvix; y is a vector in

, then

jvi  x; x  y; x ÿ y de®nes a function (a transformation) that maps

212

LINEAR VECTOR SPACES

into E

. In particular, if jvi1; 1, then the image of jvi under

jvi  1; 2; 0. It is easy to see that the transformation is linear. If

juix

; y

 and jvix

; y

, then

juijvix

 x

; y

 y

;

so that

 v

x

 x

; x

 x

y

 y

; x

 x

ÿy

 y



 x

; x

 y

; x

ÿ y

x

; x

 y

; x

ÿ y









and if k is a scalar, then

kujikx

; kx

 ky

; kx

ÿ ky

kx

; x

 y

; x

ÿ y

k

uji:

Thus

is a linear transformation.

maps the vector space onto itself (

: V ! V), then it is called a linear

operator on V.InE

a rotation of the entire space about a ®xed axis is an example

of an operation that maps the space onto itself. We saw in Chapter 3 that rotation

can be represented by a matrix with elements 

i; j  1; 2; 3;ifx

; x

are the

components of an arbitrary vector in E

before the transformation and x

; x

the components of the transformed vector, then

 

 

;

 

 

;

 

 

;

5:15

In matrix form we have





x; 5:16

where  is the angle of rotation, and



;

x 

; and





In particular, if the rotation is carried out about x

-axis,

 has the following

form:



cos  ÿsin  0

sin  cos  0

001

213

LINEAR OPERATORS

Eq. (5.16) determines the vector x

if the vector x is given, and

 is the ope rator

(matrix representation of the rotation operator) which turns x into x

Loosely speaking, an operator is any mathematical entity which operates on

any vector in V and turns it into another vector in V. Abstractly, an operator

a mapping that assigns to a vector jvi in a linear vector space V another vector jui

in V: jui

jvi. The set of vectors jvi for which the mapping is de®ned, that is,

the set of vectors jvi for which

jvi has meaning, is called the domain of

. The

set of vectors jui in the domain expressible as jui

jvi is called the range of the

operator. An operator

is linear if the mapping is such that for any vectors

jui; jwi in the domain of

and for arbitrary scalars , ÿ, the vector

juiÿjwi is in the domain of

and

juiÿjwi  

juiÿ

jwi:

A linear operator is bounded if its domain is the entire space V and if there exists a

single constant C such that

jvij < Cjjvij

for all jvi in V. We shall consider linear bounded operators only.

Matrix representation of operators

Linear bounded operators may be represented by matrix. The matrix will have a

®nite or an in®nite number of rows according to whether the dimension of V is

®nite or in®nite. To show this, let j1i; j2i; ... be an orthonormal basis in V; then

every vector j'i in V may be written in the form

j'i

j1i

j2i:

Since

ji is also in V, we may write

j'iÿ

j1iÿ

j2i:

But

j'i

j1i

j2i;

j1iÿ

j2i

j1i

j2i:

Taking the inner product of both sides with h1j we obtain

h1j

j1i

h1j

j2i

 þ



 þ



;

214

LINEAR VECTOR SPACES

Similarly

h2j

j1i

h2j

j2i

 þ



 þ



;

h3j

j1i

h3j

j2i

 þ



 þ



:

In general, we have





;

where

 i

: 5:17

Consequently, in terms of the vectors j1i; j2i; ... as a basis, operator

is repre-

sented by the matrix whose elements are þ

A matrix representing

can be found by using any basis, not necessarily an

orthonormal one. Of course, a change in the basis changes the matrix representing

The algebra of linear operators

Let

and

be two operators de®ned in a linear vector space V of vectors ji. The

equation



will be understood in the sense that

ji

ji for all ji2V:

We de®ne the addition and multiplication of linear operators as





and



if for any ji





j





;

ji

ji 



ji:

Note that



and

are themselves linear operators.

Example 5.10

(a) 

 u

 ÿ v

ji





ji

 ÿ

j i

 

 u

 ÿ

 v

;

(b)





 vji



vji

vji

ji

ji,

which shows that









215

THE ALGEBRA OF LINEAR OPERATORS