Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

282

Appendix A

1

eirs

2ð

…ÿ1

ds

rk eÿr

(A:11)

ð

(1 is)k 1

k!

…ÿðcos nè dè 2ð,

n 0

ð

0,

n 1, 2, 3, . . .

(A:12)

ð

…ÿðsin nè dè 0,

ð

n 0, 1, 2, . . .

(A:13)

…ÿð

…

ð

ðcos mè cos nè dè 2

0 cos mè cos nè dè ðämn

(A:14)

…ÿðsin mè sin

ðè dè 2…0 sin mè sin nè dè ðämn

(A:15)

n

…ÿðcos mè sin nè dè 0

(A:16)

…sin2è dè 21è ÿ 41sin 2è 21(è ÿ sin è cos è)

(A:17)

…è sin2è dè 41è2 ÿ 41è sin 2è ÿ 81 cos 2è

(A:18)

…è2 sin2è dè 61è3 ÿ 41(è2 ÿ 21)sin 2è ÿ 41è cos 2è

(A:19)

…sin3è dè 31 cos3è ÿ cos è ÿ43 cos è

1

cos 3è

(A:20)

12

…sin5è dè ÿ85 cos è

5

cos 3è ÿ

1

cos 5è

(A:21)

48

80

…sin7è dè ÿ6435 cos è

7

cos 3è ÿ

7

cos 5è

1

cos 7è

(A:22)

64

320

448

sin kè sin nè dè

sin(k ÿ n)è

sin(k n)è

,

(k2

n2)

(A:23)

2(k ÿ n)

ÿ

…

2(k n)

6

Integration by parts

…u(x)		…u dv uv ÿ …v du			(A:24)
	dv(x)	dx u(x)v(x) ÿ …v(x)	du(x)	dx	(A:25)
	dx		dx

Gamma function

1
Ã(n) …0	z nÿ1eÿz dz	(A:26)
Ã(n 1) nÃ(n)		(A:27)
Ã(n) (n ÿ 1)!,	n integer	(A:28)

Mathematical formulas

	Ã(21) ð1=2
1 x2neÿx2 dx	1z(2nÿ1)=2eÿz dz		Ã	2n 1
…ÿ1	…0			2

Trigonometric functions

eiè cos è i sin è

cos è 12(eiè eÿiè) sin è 2i1 (eiè ÿ eÿiè) sin2è cos2è 1

cos(è j) cos è cos j ÿ sin è sin j sin(è j) sin è cos j cos è sin j cos 2è cos2è ÿ sin2è

sin 2è 2sin è cos è sin 3è 3sin è ÿ 4 sin3è

sin 5è 5sin è ÿ 20 sin3è 16sin5è

sin 7è 7sin è ÿ 56sin3è 112sin5è ÿ 64sin7è

d

dè cos è ÿsin è

d

dè sin è cos è

ddz sinÿ1 z ddz arcsin z (1 ÿ z2)ÿ1=2

Hyperbolic functions

cosh è 12(eè eÿè) sinh è 12(eè ÿ eÿè) cosh iè cos è

sinh iè i sin è cosh2è ÿ sinh2è 1

tanh è sinh è cosh è

sinh 2è 2sinh è cosh è

	d cosh è			sinh è
	dè
	d sinh è			cosh è
	dè
d tanh è				1
	dè				cosh2è

283

(A:29) (A:30)

(A:31) (A:32) (A:33) (A:34) (A:35) (A:36) (A:37) (A:38) (A:39) (A:40) (A:41)

(A:42) (A:43)

(A:44)

(A:45) (A:46) (A:47) (A:48) (A:49)

(A:50) (A:51) (A:52) (A:53) (A:54)

284 Appendix A

Schwarz's inequality

j j

…ja(x)j2 dx…jb(x)j2 dx >

…a (x)b(x) dx 2

j

j

2

ÿ

j j

2

j ÿ j

2

2

For z x

iy,

z

2

> Im z

; since Im z

z

z

1

z

z =2i,

> 4 z

…ja(x)j2 dx : …jb(x)j2 dx >

[a

(x)b(x) ÿ a(x)b

41 …

(x)] dx

Vector relations

A : B AB cos è jA 3 Bj AB sin è

è is the angle between A and B

Spherical coordinates (r, è, j)

x r sin è cos j,

y r sin è sin j,

z r cos è

dô r2 sin è dr dè dj

1 @2ø

1 @

@ø

1

@

@ø

=2

ø

r2

sin è

r2

@r

@r

r2 sin è

@è

@è

r2 sin2è

@j2

Plane polar coordinates (r, j)

x r cos j,		@r	y r sin j
=2ø r @r	r			r2	@j2
1 @		@ø		1	@2ø

(A:55)

(A:56)

(A:57) (A:58)

(A:59) (A:60)

(A:61)

(A:62)

(A:63)

Appendix B

Fourier series and Fourier integral

Fourier series

An arbitrary function f (è) which satis®es the Dirichlet conditions can be expanded as

	a0		X
f (è)	2		(an cos nè bn sin nè)	(B:1)
			n 1

where è is a real variable, n is a positive integer, and the coef®cients an and bn are constants. The Dirichlet conditions specify that f (è) is single-valued, is continuous except for a ®nite number of ®nite discontinuities, and has a ®nite number of maxima and minima. The series expansion (B.1) of the function f (è) is known as a Fourier series.

We note that

cos n(è 2ð) cos nè sin n(è 2ð) sin nè

so that each term in equation (B.1) repeats itself in intervals of 2ð. Thus, the function f (è) on the left-hand side of equation (B.1) has the property

f (è 2ð) f (è)

which is to say, f (è) is periodic with period 2ð. For convenience, we select the range ÿð < è < ð for the period, although any other range of width 2ð is acceptable. If a function F(j) has period p, then it may be converted into a function f (è) with period 2ð by introducing the new variable è de®ned by è 2ðj= p, so that

f (è) F(2ðj= p). If a non-periodic function F(è) is expanded in a Fourier series, the function f (è) obtained from equation (B.1) is identical with F(è) over the range

ÿð < è < ð, but outside that range the two functions do not agree.

To ®nd the coef®cients an and bn in the Fourier series, we ®rst multiply both sides of equation (B.1) by cos mè and integrate from ÿð to ð. The resulting integrals are evaluated in equations (A.12), (A.14), and (A.16). For n 0, all the integrals on the

right-hand side vanish except the ®rst, so that

…ð f (è) dè a0 3 2ð ða0

ÿð 2

For m . 0, all the integrals on the right-hand side vanish except for the one in which n m, giving

285

286		Appendix B
		ð
		…ÿð f (è)cos mè dè ðam
If we multiply both sides of equation (B.1) by sin mè, integrate from ÿð to ð, and
apply equations (A.13), (A.15), and (A.16), we ®nd
		ð
		…ÿð f (è)sin mè dè ðbm
Thus, the coef®cients in the Fourier series are given by
	1	ð
an		…ÿð f (è)cos nè dè,	n 0, 1, 2, . . .	(B:2a)
	ð
	1	ð
bn		…ÿð f (è)sin nè dè,	n 1, 2, . . .	(B:2b)
	ð

In deriving these expressions for an and bn, we assumed that f (è) is continuous. If f (è) has a ®nite discontinuity at some angle è0 where ÿð , è0 , ð, then the expres-

sion for an in equation (B.2a) becomes
	1	è0	1	ð
an		…ÿð f (è)cos nè dè		…è0	f (è)cos nè dè
	ð		ð

A similar expression applies for bn. The generalization for a function f (è) with a ®nite number of ®nite discontinuities is straightforward. At an angle è0 of discontinuity, the Fourier series converges to a value of f (è) mid-way between the left and right values of f (è) at è0; i.e., it converges to

lim		1	[ f (è0 ÿ å) f (è0 å)]
	2
å 0
!
The Fourier expansion (B.1) may also be expressed as a cosine series or as a sine
series by the introduction of phase angles án
				a0		1
f (è)						cn cos(nè án)	(B:3a)
				2
						n 1
			1			X
					c9n sin(nè á9n)		(B:3b)
				n 0
				X

where cn, c9n, án, á9n are constants. Using equation (A.35), we may write cn cos(nè án) cn cos nè cos án ÿ cn sin nè sin án

If we let

an cn cos án bn ÿcn sin án

then equations (B.1) and (B.3a) are seen to be equivalent. Using equation (A.36), we have

c9n sin(nè á9n) c9n sin nè cos á9n c9n cos nè sin á9n

Letting
a0	2c90 sin á90
an c9n sin á9n,		n . 0
bn c9n cos á9n,		n . 0

Fourier series and Fourier integral

287

we see that equations (B.1) and (B.3b) are identical.

Other variables

The Fourier series (B.1) and (B.3) are expressed in terms of an angle è. However, in many applications the variable may be a distance x or the time t. If the Fourier series is to represent a function f (x) of the distance x in a range ÿl < x < l, we make the substitution

è

ðx

l

in equation (B.1) to give

X

a

n

x

n

x

1

ð

f (x)

0

n 1 an cos

ð

bn sin

(B:4)

2

l

l

with an and bn given by

1

l

n

x

an

…ÿl

f (x)cos

ð

dx,

n 0, 1, 2, . . .

(B:5a)

l

l

1

l

n

ð

x

bn

…ÿl f (x)sin

dx,

n 1, 2, . . .

(B:5b)

l

l

If time is the variable, then we may make either of the substitutions

è

2ðt

ùt

p

where p is the period of the function f (t) and ù is the angular frequency, so that

equation (B.1) becomes

X

a

X

2

nt

2

nt

a

1

1

f (t)

0

n

1 an cos

ð

bn sin

ð

0

n

(an cos nùt bn sin nùt)

2

p

p

2

1

(B:6)

The constants an and bn in equations (B.2) for the variable t are

2

p=2

2

nt

ù

ð=ù

an

…ÿ p=2 f (t)cos

ð

dt

…ÿð=ù f (t)cos nùt dt

(B:7a)

p

p

ð

2

p=2

2

nt

ù

ð=ù

bn

…ÿ p=2 f (t)sin

ð

dt

…ÿð=ù f (t)sin nùt dt

(B:7b)

p

p

ð

Complex form

The Fourier series (B.1) can also be written in complex form by the substitution of equations (A.32) and (A.33) for cos nè and sin nè, respectively, to yield

X1

f (è)	cn einè	(B:8)
	n ÿ1

where

	288				Appendix B
		c	n		an ÿ ibn		,	n . 0
					2
	c	ÿn			an ibn		,	n . 0	(B:9)
					2
		c0			a0
					2

The coef®cients cn in equation (B.8) may be obtained from (B.9) with an and bn given by (B.2). The result is

		1	ð
	cn		…ÿð f (è)eÿinè dè	(B:10)
		2ð

which applies to all values of n, positive and negative, including n 0. We note in

passing that c	ÿ	n is the complex conjugate cn					of cn.
In terms of the distance variable x, equations (B.8) and (B.10) become
						1	cn einðx=l
		f (x)						(B:11)
						n ÿ1
						X
		1				l
		cn				…ÿl f (x)eÿinðx=l dx		(B:12)
				2l
while in terms of the time t, they take the form
						1	cn einùt
		f (t)						(B:13)
						n ÿ1
						X
		ù				ð=ù
		cn				…ÿð=ù f (t)eÿinùt dt		(B:14)
			2ð

Parseval's theorem

We now investigate the relation between the average of the square of f (è) and the coef®cients in the Fourier series for f (è). For this purpose we select the Fourier series (B.8), although any of the other expansions would serve as well. In this case the average of jf (è)j2 over the interval ÿð < è < ð is

1

ð

…ÿðjf (è)j2 dè

2ð

The square of the absolute value of f (è) in equation (B.8) is

n

1

2

1

1

j

f (è) 2

cn einè

m

cm cn ei(nÿm)è

(B:15)

n

j

ÿ1

X

X

X

where the two independent summations

have been assigned different dummy indices.

Integration of both sides of equation (B.15) over è from ÿð to ð gives

ð

1

1

ð

…ÿðj

f (è) 2

dè

m

n

cm cn

…ÿð

ei(nÿm)è dè

j

X

X

ÿ1 ÿ1

Since m and n are integers, the integral on the right-hand side vanishes except when m n, so that we have

Fourier series and Fourier integral					289
		ð
The ®nal result is		…ÿðei(nÿm)è dè 2ðämn
			X
1		ð
				1
		…ÿðjf (è)j2 dè n		jcnj2	(B:16)
2ð
				ÿ1

which is one form of Parseval's theorem. Other forms of Parseval's theorem are obtained using the various alternative Fourier expansions.

Parseval's theorem is also known as the completeness relation and may be used to verify that the set of functions ei nè for ÿ1 < n < 1 are complete, as discussed in Section 3.4. If some of the terms in the Fourier series are missing, so that the set of basis functions in the expansion is incomplete, then the corresponding coef®cients on the right-hand side of equation (B.16) will also be missing and the equality will not hold.

Fourier integral

The Fourier series expansions of a function f (x) of the variable x over the range ÿl < x < l may be generalized to the case where the range is in®nite, i.e., where

ÿ1 < x < 1. By a suitable limiting process in which l ! 1, equations (B.11) and (B.12) may be extended to the form

	1	1
f (x) p2ð		…ÿ1 g(k)eikx dk		(B:17)
g(k)	1••••••	1	f (x)eÿi kx dx	(B:18)
	p2ð	…ÿ1		g(k) is the
Equation (B.17) is the Fourier integral••••••

representation of f (x). The function

Fourier transform of f (x), which in turn is the inverse Fourier transform of g(k). For any function f (x) which satis®es the Dirichlet conditions over the range

ÿ1 < x < 1 and for which the integral

…1

jf (x)j2 dx

ÿ1

converges, the Fourier integral in equation (B.17) converges to f (x) wherever f (x) is continuous and to the mean value of f (x) at any point of discontinuity.

In some applications a function f (x, t), where x is a distance variable and t is the

time, is represented as a Fourier integral of the form
	1			1
f (x, t) p2ð			…ÿ1G(k)ei[kxÿù(k)t] dk			(B:19)
where the frequency ù(k) depends	••••••				k. In this case the Fourier transform
	on the variable
g(k) takes the form				G(k)eÿiù(k)t
g(k)
and equation (B.18) may be written as
1		1				(B:20)
G(k) p2ð …ÿ1 f (x, t)eÿi[kxÿù(k)t] dx
••••••

The functions f (x, t) and G(k) are, then, a generalized form of Fourier transforms.

290		Appendix B
Another generalized form may be obtained by exchanging the roles of x and t in
equations (B.19) and (B.20), so that
	1	1
f (x,	t) p2ð	…ÿ1G(ù)ei[k(ù)xÿùt] dù		(B:21)
G(ù)	1 ••••••	1	f (x, t)eÿi[k(ù)xÿùt] dt	(B:22)
	p2ð …ÿ1
	••••••

Fourier integral in three dimensions

The Fourier integral may be readily extended to functions of more than one variable. We now derive the result for a function f (x, y, z) of the three spatial variables x, y, z. If we consider f (x, y, z) as a function only of x, with y and z as parameters, then we have

1

1

f (x, y,

z) p2ð

…ÿ1 g1(kx,

y, z)ei kx x dkx

(B:23a)

1

1

ikx x

••••••

We next regard g

g1(kx, y, z) p2ð …ÿ1 f (x, y, z)eÿ

dx

(B:23b)

1(kx, y, z) as a

function only of y with k

x

and z as parameters and

••••••

express g1(kx, y, z) as a Fourier integral

1

1

g1(kx,

y,

z) p2ð

…ÿ1 g2(kx, ky, z)eik y y dky

(B:24a)

1

1

ik y y

••••••

Considering g

g2(kx, ky, z) p2ð …ÿ1 g1(kx,

y, z)eÿ

d y

(B:24b)

2(kx, ky, z) as a

function only of z, we have

••••••

1

1

g2(kx, ky,

z) p2ð

…ÿ1 g(kx, ky, kz)eikz z dkz

(B:25a)

1

1

g2(kx, ky, z)eÿikz z dz

g(kx, ky, kz)

••••••

(B:25b)

p2ð …ÿ1

Combining equations (B.23a), (B.24a), and (B.25a), we obtain

1

••••••

f (x, y, z)

1

……… g(kx, ky, kk )ei(kx x k y y kz z) dkx dky dkz

(B:26a)

(2ð)3=2

ÿ1

Combining equations (B.23b), (B.24b), and (B.25b), we have

1

g(kx, ky, k k )

1

……… f (x, y, z)eÿi(kx x k y y kz z) dx d y dz

(B:26b)

(2ð)3=2

ÿ1

If we de®ne the vector r with components x, y, z and the vector k with components kx, ky, kz and write the volume elements as

dr dx dy dz

dk dkx dky dkz

then equations (B.26) become

f (x) f (x) dx 21ð

Fourier series and Fourier integral					291
f (r)		1		… g(k)eik.r dk	(B:27a)
		(2ð)3=2
g(k)	1			… f (r)eÿik.r dr	(B:27b)
	(2ð)3=2

Parseval's theorem

To obtain Parseval's theorem for the function f (x) in equation (B.17), we ®rst take the

complex conjugate of f (x)
	1	1
f (x) p2ð		…ÿ1 g (k9)eÿik9x dk9
where we have used a different	dummy variable of integration. The integral of the
	••••••
square of the absolute value of f (x) is then given by
…1 jf (x)j2 dx …1		1
				…… g (k9) g(k)ei(kÿk9)x dk dk9 dx

ÿ1 ÿ1

ÿ1

The order of integration on the right-hand side may be interchanged. If we integrate

over x while noting that according to equation (C.6)

…1

	ei(kÿk9)x dx				2ðä(k	ÿ	k9)
	ÿ1
we obtain
	…ÿ1jf (x)j2 dx	…… g (k9)g(k)ä(k ÿ k9) dk dk9
	1	1
		ÿ1
Finally, integration over the variable k9 yields Parseval's theorem for the Fourier
integral,	1				1
	…ÿ1jf (x)j2 dx			…ÿ1jg(k)j2 dk				(B:28)
Parseval's theorem for the functions f (r) and g(k) in equations (B.27) is
	…jf (r)j2 dr			…jg(k)j2 dk				(B:29)

This relation may be obtained by the same derivation as that leading to equation (B.28), using the integral representation (C.7) for the three-dimensional Dirac delta function.