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

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4.10 Find the exponential form of the Fourier series of the function whose de®-

nition in one period is f xe

ÿx

; ÿ1 < x < 1.

4.11 (a) Show that the set of functions

1; sin

x

; cos

x

; sin

2x

; cos

2x

; sin

3x

; cos

3x

; ...

form an orthogonal set in the interval ÿL; L.

(b) Determine the corresponding normalizing constants for the set in (a)so

that the set is orthonormal in ÿL; L.

4.12 Express f x; yxy as a Fourier series for 0  x  1; 0  y  2.

4.13 Steady-state heat conduction in a rectangular plate: Consider steady-state

heat conduction in a ¯at plate having temperature values prescribed on the

sides (Fig. 4.22). The boundary value problem modeling this is:



 0; 0 < x <; 0 < y <ÿ;

ux; 0ux;ÿ0; 0 < x <;

u0; y0; u; yT; 0 < y <ÿ:

Determine the temperature at any point of the plate.

4.14 Derive and solve the following eigenvalue problem which occurs in the

theory of a vibrating square membrane whose sides, of length L, are kept

®xed:



 w  0;

w0; ywL; y0 0  y  L;

wx; 0wx; L0 0  y  L:

196

FOURIER SERIES AND INTEGRALS

Figure 4.22. Flat plate with prescribed temperature.

4.15 Show that the Fourier integral can be written in the form

f x



ÿ1

f x

cos !x ÿ x

dx

4.16 Starting with the form obtained in Problem 4.15, show that the Fourier

integral can be wri tten in the form

f x

A!cos !x  B!sin !x

d!;

where

A!



ÿ1

f xcos !xdx; B ! 



ÿ1

f xsin !xdx:

4.17 (a) Find the Fourier transform of

f x

1 ÿ x

jxj < 1

0 jxj > 1

(

(b) Evaluate

x cos x ÿ sin x

cos

dx:

4.18 (a) Find the Fourier cosine transform of f xe

ÿmx

; m > 0.

(b) Use the result in (a) to show that

cos px

 

dx 



2

ÿp

p > 0;>0 :

4.19 Solve the integral equation

f xsin xdx

1 ÿ  0    1

0 >1



4.20 Find a bounde d solution to Laplace's equation r

ux; y0 for the half-

plane y > 0ifu takes on the value of f (x) on the x-axis:



 0; ux; 0f x; ux; y

< M:

4.21 Show that the following two functions are valid representations of the delta

function, where " is positive and real:

a x





lim

"!0



ÿx

b x



lim

"!0

 "

197

PROBLEMS

4.22 Verify the following properties of the delta function:

(a) xÿx,

(b) xx0,

ÿxÿ

x,

(d) x

xÿx,

(e) ccxx; c > 0.

4.23 Solve the integral equation for yx

ÿ1

yudu

x ÿ u

 a



 b

0 < a < b:

4.24 Use Fourier transforms to solve the boundary value problem

 k

; ux; 0f  x; u x; t

< M;

where ÿ1 < x < 1; t > 0.

4.25 Obtain a solution to the equation of a driven harmonic oscillator



xt2ÿ

xt!

xt0  Rt;

where ÿ and !

are positive and real constants.

198

FOURIER SERIES AND INTEGRALS

Linear vector spaces

Linear vector space is to quantum mechanics what calculus is to classical

mechanics. In this chapter the essential ideas of linear vector spaces will be dis-

cussed. The reader is already familiar with vector calculus in three-dimensional

Euclidean space E

(Chapter 1). We therefore present our discussion as a general -

ization of elementary vector calculus. The present ation will be, however, slightly

abstract and more form al than the discussion of vectors in Chapter 1. Any reader

who is not already familiar with this sort of discussion should be patient with the

®rst few sections. You will then be amply repaid by ®nding the rest of this chapter

relatively easy reading.

Euclidean n-space E

In the study of vector analysis in E

, an ordered triple of numbers (a

, a

) has

two diÿerent geometric interpretations. It represents a point in space, with a

, a

being its coordinates; it also represents a vector, with a

, a

, and a

being its

components along the three coordinate axes (Fig. 5.1). This idea of using triples of

numbers to locate points in three-dimensional space was ®rst introduced in the

mid-seventeenth century. By the latter part of the nineteenth century physicists

and mathematicians began to use the quadruples of numbers ( a

, a

)as

points in four-dimensional space, quintuples (a

, a

) as points in ®ve-

dimensional space etc. We now extend this to n-dimensional space E

, where n is a

positive integer. Although our geometric visualization doesn't extend beyond

three-dimensional space, we can extend many familiar ideas beyond three-dimen-

sional space by working with analytic or numerical properties of points and

vectors rather than their geometric properties.

For two- or three-dimensional space, we use the terms `ordered pair' and

`ordered triple.' When n > 3, we use the term `ordered-n-tuplet' for a sequence

199

of n numbers, real or complex, (a

, a

; ...; a

); they will be viewed either as a

generalized point or a generalized vector in a n-dimensional space E

Two vectors u u

; u

; ...; u

 and v v

; v

; ...; v

 in E

are called equal if

 v

; i  1; 2; ...; n 5:1

The sum u  v is de®ned by

u  v u

 v

; u

 v

; ...; u

 v

5:2

and if k is any scalar, the scalar multiple ku is de®ned by

ku ku

; ku

; ...; ku

: 5:3

If u u

; u

; ...; u

 is any vector in E

, its negative is given by

ÿu ÿu

; ÿu

; ...; ÿu

5:4

and the subtraction of vectors in E

can be considered as addition: v ÿ u 

v ÿu. The null (zero) vector in E

is de® ned to be the vector 0 0; 0; ...; 0.

The addition and scalar multiplication of vectors in E

have the following

arithmetic properties:

u  v  v  u; 5:5a

u v  wu  vw; 5:5b

u  0  0  u  u; 5:5c

abuabu; 5:5d

au  vau  av; 5:5e

a  bu  au  bu; 5:5f

where u, v, w are vectors in E

and a and b are scalars.

200

LINEAR VECTOR SPACES

Figure 5.1. A space point P whose position vector is A.

We usually de®ne the inner product of two vectors in E

in terms of lengths of

the vectors and the angle between the vectors: A  B  AB cos ;  þA; B.We

do not de®ne the inner product in E

in the same manner. However, the inner

product in E

has a second equivalent expression in terms of comp onents:

A  B  A

 A

. We choose to de®ne a similar formula for the gen-

eral case. We made this choice because of the further generalization that will be

outlined in the next section. Thus, for any two vectors u u

; u

; ...; u

 and

v v

; v

; ...; v

 in E

, the inner (or dot) product u  v is de®ned by

u  v  u

 u

u

5:6

where the asterisk denotes complex conjugation. u is often called the prefactor

and v the post-factor. The inner product is linear with respect to the post-factor,

and anti-linear with respect to the prefactor:

u av  bwau  v  bu  w; au  bvw  a*u  vb*u  w:

We expect the inner product for the general case also to have the following three

main features:

u  v v  u* 5:7a

u av  bwau  v  bu  w 5:7b

u  u  0  0; if and only if u  0: 5:7c

Many of the familiar ideas from E

and E

have been carried over, so it is

common to refer to E

with the operations of addition, scalar multiplication, and

with the inner product that we have de®ned here as Euclidean n-space.

General linear vector spaces

We now generalize the concept of vector space still further: a set of `objects' (or

elements) obeying a set of axioms, which will be chosen by abstracting the most

important properties of vectors in E

, forms a linear vector space V

with the

objects called vectors. Before introducing the requisite axioms, we ®rst adapt a

notation for our general vectors: general vectors are designa ted by the symbol ji,

which we call, following Dirac, ket vectors; the conjugates of ket vectors are

denoted by the symbol hj, the bra vectors. However, for simplicity, we shall

refer in the future to the ket vectors jisimply as vectors, and to the hjsas

conjugate vectors. We now proceed to de®ne two basic operations on these

vectors: addition and multiplication by scalars.

By addition we mean a rule for forming the sum, denoted jý

ijý

i, for

any pair of vectors jý

i and jý

By scalar multiplication we mean a rule for associating with each scalar k

and each vector jýi a new vector kjýi.

201

GENERAL LINEAR VECTOR SPACES

We now proceed to generalize the concept of a vector space. An arbitrary set of

n objects j1i; j2i; j3i; ...; ji; ...; j'i form a linear vector V

if these objects, called

vectors, meet the following axioms or properties:

A.1 If 

and '

are objects in V

and k is a scalar, then 

 '

and k 

are

in V

, a feature called closure.

A.2 

 '

 '

 

; that is, addition is commut ative.

A.3 ( 

 '

ji

 ý

 

'

 ý

); that is, addition is associative.

A.4 k 

 '

ji

 k 

 k '

; that is, scalar multiplication is distributive in the

vectors.

A.5 k   

 k 

 

; that is, scalar multiplication is distributive in the

scalars.

A.6 k

ji

 k

; that is, scalar multiplication is associative .

A.7 There exists a null vector 0

in V

such that 

 0

 

for all 

in V

A.8 For every vector 

in V

, there exists an inverse under addition, ÿ

such

that 

ÿ

 0

The set of numbers a; b; ... used in scalar multiplication of vectors is called the

®eld over which the vector ®eld is de®ned. If the ®eld consists of real numbers, we

have a real vector ®eld; if they are complex, we have a complex ®eld. Note that the

vectors themselves are neither real nor complex, the nature of the vectors is not

speci®ed. Vectors can be any kinds of objects; all that is required is that the vector

space axioms be satis®ed. Thus we purposely do not use the symbol V to denote

the vectors as the ®rst step to turn the reader away from the limited concept of the

vector as a directed line segment. Instead, we use Dirac's ket and bra symbols,

and j

, to denote generic vectors.

The familiar three-dimensional space of position vectors E

is an example of

a vector space over the ®eld of real numbers. Let us now examine two simple

examples.

Example 5.1

Let V be any plane through the origin in E

. We wish to show that the points in

the plane V form a vector space under the addition and scalar multiplication

operations for vector in E

Solution: Since E

itself is a vector space under the addition and scalar multi-

plication operations, thus Axioms A.2, A.3, A.4, A.5, and A.6 hold for all points

in E

and consequently for all points in the plane V. We therefore need only show

that Axioms A.1, A.7, and A.8 are satis®ed.

Now the plane V, passing through the origin, has an equation of the form

 bx

 cx

 0:

202

LINEAR VECTOR SPACES

Hence, if u u

; u

 and v v

; v

 are points in V, then we have

 bu

 cu

 0andav

 bv

 cv

 0:

Addition gives

au

 v

bu

 v

cu

 v

0;

which shows that the point u  v also lies in the plane V. This proves that Axiom

A.1 is satis®ed. Multiplying au

 bu

 cu

 0 through by ÿ1 gives

aÿu

bÿu

cÿu

0;

that is, the point ÿu ÿu

; ÿu

 lies in V. This establishes Axiom A.8.

The veri®cation of Axiom A.7 is left as an exercise.

Example 5.2

Let V be the set of all m  n matrices with real elements. We know how to add

matrices and multiply matrices by scala rs. The corresponding rules obey closure,

associativity and distributive requirements. The null matrix has all zeros in it, and

the inverse under matrix addition is the matrix with all elements negated. Thus the

set of all m  n matrices, together with the operations of matrix addition and

scalar multiplication, is a vector space. We shall denote this vector space by the

symbol M

Subspaces

Consider a vector space V.IfW is a subset of V and form s a vector space under

the addition and scalar multiplication, then W is called a subspace of V. For

example, lines and planes passing through the origin form vector spaces and

they are subspaces of E

Example 5.3

We can show that the set of all 2  2 matrices having zero on the main diagonal is

a subspace of the vector space M

of all 2  2 matrices.

Solution: To prove this, let

X 

0 x

ý!

Y 

0 y

ý!

be two matrices in W and k any scalar. Then

X 

0 x

ý!

and

X 

Y 

0 x

 y

ý!

and thus they lie in W. We leave the veri®cation of other axioms as exercises.

203

SUBSPACES

Linear combination

A vector W

is a linear combination of the vectors v

; v

; ...; v

if it can be

expressed in the form

 k

ik

ik

where k

; k

; ...; k

are scalars. For example, it is easy to show that the vector

9; 2; 7  in E

is a linear combination of v

1; 2; ÿ1 and

6; 4; 2. To see this, let us write

9; 2; 7k

1; 2; ÿ1k

6; 4; 2

9; 2; 7k

 6k

; 2k

 4k

; ÿk

 2k

:

Equating corresponding components gives

 6k

 9; 2k

 4k

 2; ÿk

 2k

 7:

Solving this system yields k

ÿ3 and k

 2 so that

ÿ3 v

 2 v

Linear independence, bases, and dimensionality

Consider a set of vectors 1

; 2

; ...; r

; ... n

in a linear vector space V. If every

vector in V is expressible as a linear combination of 1

; 2

; ...; r

; ...; n

, then

we say that these vectors span the vector space V, and they are call ed the base

vectors or basis of the vector space V. For example, the three unit vectors

1; 0; 0; e

0; 1; 0 , and e

0; 0; 1 span E

because every vector in E

is expressible as a linear combination of e

, e

, and e

. But the following three

vectors in E

do not span E

: 1

1; 1; 2 ; 2

1; 0; 1, and 3

2; 1; 3.

Base vectors are very useful in a variety of problems since it is often possible to

study a vector space by ®rst studying the vectors in a base set, then extending the

results to the rest of the vector space. Therefore it is desirable to keep the spanning

set as small as possible. Finding the spanning sets for a vector space depends upon

the notion of linear independence.

We say that a ®nite set of n vectors 1

; 2

; ...; r

; ...; n

, none of which is a

null vector, is linearly independent if no set of non-zero numbers a

exists such

that

k1

j0i: 5:8

In other words, the set of vectors is linearly independent if it is impossible to

construct the null vector from a linear combination of the vectors except when all

204

LINEAR VECTOR SPACES

the coecients vanish. For example, non-zero vectors 1jiand 2jiof E

that lie

along the same coordinate axis, say x

, are not linearly independent, since we can

write one as a mult iple of the other: 1

 a 2

, where a is a scalar which may be

positive or negative. That is, 1

and 2

depend on each other and so they are not

linearly independent. Now let us move the term a 2

to the left hand side and the

result is the null vector: 1

ÿ a 2

 0

. Thus, for these two vectors 1

and 2

, we can ®nd two non-zero numbers (1, ÿa such that Eq. (5.8) is satis®ed, and

so they are not linearly independent.

On the other hand, the n vectors 1

; 2

; ...; r

; ...; n

are linearly dependent if

it is possible to ®nd scalars a

; a

; ...; a

, at least two of which are non-zero, such

that Eq. (5.8) is satis®ed. Let us say a

6 0. Then we could express 9

in terms of

the other vectors



i1;69

ÿa

That is, the n vectors in the set are linearly dependent if any one of them can be

expressed as a linea r combination of the remaining n ÿ 1 vectors.

Example 5.4

The set of three vectors 1

2; ÿ1; 0; 3,2

1; 2; 5; ÿ1; 3

7; ÿ1; 5; 8 is

linearly dependent, since 3 1ji2jiÿ3ji0ji.

Example 5.5

The set of three unit vectors e

1; 0; 0 ; e

0; 1; 0, and e

0; 0; 1 in

is linearly independent. To see this, let us star t with Eq. (5.8) which now takes

the form

 a

 0

1; 0; 0a

0; 1; 0a

0; 0; 10; 0; 0

from which we obtain

a

; a

0; 0; 0 ;

the set of three unit vectors e

; e

, and e

is therefore linearly independent.

Example 5.6

The set S of the following four matrices





; 2





; j3





; 4





;

205

LINEAR INDEPENDENCE, BASES, AND DIMENSIONALITY