Page 299 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

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 i[k(ù)xÿùt]
f (x, t) ˆ p  G(ù)e dù (B:21)
2ð ÿ1

1 1 ÿi[k(ù)xÿùt]
G(ù) ˆ p  f (x, t)e dt (B:22)
2ð 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 ik x x
f (x, y, z) ˆ p  g 1 (k x , y, z)e dk x (B:23a)
2ð ÿ1
1 1
g 1 (k x , y, z) ˆ p  f (x, y, z)e ÿik x x dx (B:23b)
2ð ÿ1
We next regard g 1 (k x , y, z) as a function only of y with k x and z as parameters and
express g 1 (k x , y, z) as a Fourier integral
1 1 ik y y
g 1 (k x , y, z) ˆ p  g 2 (k x , k y , z)e dk y (B:24a)
2ð ÿ1
1 1 ÿik y y
g 2 (k x , k y , z) ˆ p  g 1 (k x , y, z)e dy (B:24b)
2ð ÿ1
Considering g 2 (k x , k y , z) as a function only of z,wehave

1 1 ik z z
g 2 (k x , k y , z) ˆ p  g(k x , k y , k z )e dk z (B:25a)
2ð ÿ1

1 1 ÿik z z
g(k x , k y , k z ) ˆ p  g 2 (k x , k y , z)e dz (B:25b)
2ð ÿ1
Combining equations (B.23a), (B.24a), and (B.25a), we obtain
1

1 i(k x x‡k y y‡k z z)
f (x, y, z) ˆ 3=2 g(k x , k y , k k )e dk x dk y dk z (B:26a)
(2ð)
ÿ1
Combining equations (B.23b), (B.24b), and (B.25b), we have
1

1 ÿi(k x x‡k y y‡k z z)
g(k x , k y , k k ) ˆ f (x, y, z)e dx dy 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
k x , k y , k z and write the volume elements as
dr ˆ dx dy dz
dk ˆ dk x dk y dk z
then equations (B.26) become