Page 49 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

40 Schro Èdinger wave mechanics

so that

@ 2 i" @ @Ø @Ø
jØ(x, t)j ˆ Ø ÿ Ø (2:12)
@t 2m @x @x @x
Substitution of equation (2.12) into (2.10) and evaluation of the integral give
1

dN i" 1 @ @Ø @Ø i" @Ø @Ø
ˆ Ø ÿ Ø dx ˆ Ø ÿ Ø
dt 2m @x @x @x 2m @x @x
ÿ1 ÿ1
Since Ø(x, t) goes to zero as x goes to ( ) in®nity, the right-most term
vanishes and we have
dN
ˆ 0
dt
Thus, the integral N is time-independent and the normalization of Ø(x, t) does
not change with time.
Not all wave functions can be normalized. In such cases the quantity
2
jØ(x, t)j may be regarded as the relative probability density, so that the ratio

a 2
2
jØ(x, t)j dx
a 1

b 2
2
jØ(x, t)j dx
b 1
represents the probability that the particle will be found between a 1 and a 2
relative to the probability that it will be found between b 1 and b 2 .Asan
example, the plane wave
Ø(x, t) ˆ e i( pxÿEt)="
does not approach zero as x approaches ( ) in®nity and consequently cannot
2
be normalized. The probability density jØ(x, t)j is unity everywhere, so that
the particle is equally likely to be found in any region of a speci®ed width.

Momentum-space wave function
The wave function Ø(x, t) may be represented as a Fourier integral, as shown
in equation (2.7), with its Fourier transform A(p, t) given by equation (2.8).
The transform A( p, t) is uniquely determined by Ø(x, t) and the wave function
Ø(x, t) is uniquely determined by A( p, t). Thus, knowledge of one of these
functions is equivalent to knowledge of the other. Since the wave function
Ø(x, t) completely describes the physical system that it represents, its Fourier
transform A(p, t) also possesses that property. Either function may serve as a
complete description of the state of the system. As a consequence, we may
2
interpret the quantity jA( p, t)j as the probability density for the momentum at