Page 20 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.2 Wave packet 11

Figure 1.5 shows the real part of the plane wave exp[i(k 0 x ÿ ù 0 t)] with its
amplitude modulated by B(x, t) of equation (1.20). The plane wave moves in
the positive x-direction with phase velocity v ph equal to ù 0 =k 0 . The maximum
amplitude occurs at x ˆ v g t and propagates in the positive x-direction with
group velocity v g equal to (dù=dk) 0 .
p 
The value of the function A(k) falls from its maximum value of ( 2ðá) ÿ1 at
p 
k 0 to 1=e of its maximum value when jk ÿ k 0 j equals 2á. Most of the area
under the curve (actually 84.3%) comes from the range
p  p 
ÿ 2á , (k ÿ k 0 ) , 2á
p 
Thus, the distance 2á may be regarded as a measure of the width of the
distribution A(k) and is called the half width. The half width may be de®ned
using 1=2 or some other fraction instead of 1=e. The reason for using 1=eis
that the value of k at that point is easily obtained without consulting a table of
numerical values. These various possible de®nitions give different numerical
values for the half width, but all these values are of the same order of
magnitude. Since the value of jØ(x, t)j falls from its maximum value of
p  p 
(2ð) ÿ1=2 to 1=e of that value when jx ÿ v g tj equals 2=á, the distance 2=á
may be considered the half width of the wave packet.
When the parameter á is small, the maximum of the function A(k) is high
and the function drops off in value rapidly on each side of k 0 , giving a small
value for the half width. The half width of the wave packet, however, is large
because it is proportional to 1=á. On the other hand, when the parameter á is
large, the maximum of A(k) is low and the function drops off slowly, giving a
large half width. In this case, the half width of the wave packet becomes small.
If we regard the uncertainty Äk in the value of k as the half width of the
distribution A(k) and the uncertainty Äx in the position of the wave packet as
its half width, then the product of these two uncertainties is
ÄxÄk ˆ 2

x

Figure 1.5 The real part of a wave packet for a gaussian wave number distribution.