Page 46 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.2 The wave function 37
@Ø ÿi 1 i( pxÿEt)="
ˆ p  EA( p)e dp (2:5)
@t 2ð" 3 ÿ1
The total energy E for a free particle (i.e., for a particle moving in a region of
constant potential energy V)isgiven by
p 2
E ˆ ‡ V
2m
which may be combined with equations (2.4) and (2.5) to give
2
2
@Ø " @ Ø
i" ˆÿ ‡ VØ
@t 2m @x 2
Schrodinger (1926) postulated that this differential equation is also valid
È
when the potential energy is not constant, but is a function of position. In that
case the partial differential equation becomes
2
2
@Ø(x, t) " @ Ø(x, t)
i" ˆÿ ‡ V(x)Ø(x, t) (2:6)
@t 2m @x 2
which is known as the time-dependent Schro Èdinger equation. The solutions
Ø(x, t) of equation (2.6) are the time-dependent wave functions. An important
goal in wave mechanics is solving equation (2.6) for Ø(x, t) using various
expressions for V(x) that relate to speci®c physical systems.
When V(x) is not constant, the solutions Ø(x, t) to equation (2.6) may still
be expanded in the form of a wave packet,

1 1 i( pxÿEt)="
Ø(x, t) ˆ p   A(p, t)e dp (2:7)
2ð" ÿ1
The Fourier transform A(p, t) is then, in general, a function of both p and time
t, and is given by

1 1 ÿi( pxÿEt)="
A( p, t) ˆ p   Ø(x, t)e dx (2:8)
2ð" ÿ1
By way of contrast, recall that in treating the free particle as a wave packet in
Chapter 1, we required that the weighting factor A( p) be independent of time
and we needed to specify a functional form for A( p) in order to study some of
the properties of the wave packet.

2.2 The wave function
Interpretation
È
Before discussing the methods for solving the Schrodinger equation for speci®c
choices of V(x), we consider the meaning of the wave function. Since the wave
function Ø(x, t) is identi®ed with a particle, we need to establish the connec-
tion between Ø(x, t) and the observable properties of the particle. As in the