Page 48 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.2 The wave function 39

A function that obeys this equation is said to be normalized. If a function
Ö(x, t) is not normalized, but satis®es the relation

1

Ö (x, t)Ö(x, t)dx ˆ N
ÿ1
then the function Ø(x, t) de®ned by
1
Ø(x, t) ˆ p  Ö(x, t)
N
is normalized.
In order for Ø(x, t) to satisfy equation (2.9), the wave function must be
square-integrable (also called quadratically integrable). Therefore, Ø(x, t)
p 
must go to zero faster than 1= jxj as x approaches ( ) in®nity. Likewise, the
derivative @Ø=@x must also go to zero as x approaches ( ) in®nity.
Once a wave function Ø(x, t) has been normalized, it remains normalized as
time progresses. To prove this assertion, we consider the integral

1

N ˆ Ø Ø dx
ÿ1
and show that N is independent of time for every function Ø that obeys the
È
Schrodinger equation (2.6). The time derivative of N is
1
dN @ 2
ˆ jØ(x, t)j dx (2:10)
dt @t
ÿ1
where the order of differentiation and integration has been interchanged on the
right-hand side. The derivative of the probability density may be expanded as
follows
@ @ @Ø @Ø
2

jØ(x, t)j ˆ (Ø Ø) ˆ Ø ‡ Ø
@t @t @t @t
Equation (2.6) and its complex conjugate may be written in the form
2
@Ø i" @ Ø i
ˆ ÿ VØ
@t 2m @x 2 "
(2:11)
2
@Ø i" @ Ø i
ˆÿ ‡ VØ
@t 2m @x 2 "
2
so that @jØ(x, t)j =@t becomes

2
2
@ i" @ Ø @ Ø
2
jØ(x, t)j ˆ Ø ÿ Ø
@t 2m @x 2 @x 2
where the terms containing V cancel. We next note that

2
2
@ @Ø @Ø @ Ø @ Ø
Ø ÿ Ø ˆ Ø ÿ Ø
@x @x @x @x 2 @x 2