Page 29 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

20 The wave function
p 2
E ˆ (1:36)
2m
Following the theoretical scheme of Schrodinger, we associate a wave packet
È
Ø(x, t) with the motion in the x-direction of this free particle. This wave
packet is readily constructed from equation (1.11) by substituting (1.32) and
(1.33) for ù and k, respectively

1 1 i( pxÿEt)="
Ø(x, t) ˆ p   A( p)e dp (1:37)
2ð" ÿ1
where, for the sake of symmetry between Ø(x, t) and A( p), a factor " ÿ1=2 has
been absorbed into A(p). The function A(k) in equation (1.12) is now
" 1=2 A( p), so that
1 1 ÿi( pxÿEt)="
A(p) ˆ p   Ø(x, t)e dx (1:38)
2ð" ÿ1
The law of dispersion for this wave packet may be obtained by combining
equations (1.32), (1.33), and (1.36) to give
E p 2 "k 2
ù(k) ˆ ˆ ˆ (1:39)
" 2m" 2m
2
This dispersion law with ù proportional to k is different from that for
undispersed light waves, where ù is proportional to k.
If ù(k) in equation (1.39) is expressed as a power series in k ÿ k 0 , we obtain
"k 2 0 "k 0 " 2
ù(k) ˆ ‡ (k ÿ k 0 ) ‡ (k ÿ k 0 ) (1:40)
2m m 2m
This expansion is exact; there are no terms of higher order than quadratic.
From equation (1.40) we see that the phase velocity v ph of the wave packet is
given by
ù 0 "k 0
v ph ˆ ˆ (1:41)
k 0 2m
and the group velocity v g is

dù "k 0
v g ˆ ˆ (1:42)
dk 0 m
while the parameter ã of equations (1.28), (1.30), and (1.31) is

1 d ù "
2
㠈 ˆ (1:43)
2 dk 2 0 2m
If we take the derivative of ù(k) in equation (1.39) with respect to k and use
equation (1.33), we obtain
dù "k p
ˆ ˆ ˆ v
dk m m