Page 21 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

12 The wave function

Thus, the product of these two uncertainties Äx and Äk is a constant of order
unity, independent of the parameter á.

Square pulse wave number distribution
As a second example, we choose A(k) to have a constant value of unity for k
between k 1 and k 2 and to vanish elsewhere, so that
A(k) ˆ 1, k 1 < k < k 2
(1:21)
ˆ 0, k , k 1 , k . k 2
as illustrated in Figure 1.6(a). With this choice for A(k), the modulating
function B(x, t) in equation (1.17) becomes

1 k 2 i(xÿv g t)(kÿk 0 )
B(x, t) ˆ p  e dk
2ð k 1
1
ˆ p  [e i(xÿv g t)(k 2 ÿk 0 ) ÿ e i(xÿv g t)(k 1 ÿk 0 ) ]
2ði(x ÿ v g t)
1
ˆ p  [e i(xÿv g t)Äk=2 ÿ e ÿi(xÿv g t)Äk=2 ]
2ði(x ÿ v g t)

r 
2 sin[(x ÿ v g t)Äk=2]
ˆ (1:22)
ð x ÿ v g t
where k 0 is chosen to be (k 1 ‡ k 2 )=2, Äk is de®ned as (k 2 ÿ k 1 ), and equation
(A.33) has been used. The function B(x, t) is shown in Figure 1.6(b).
The real part of the wave packet Ø(x, t) obtained from combining equations
(1.14) and (1.22) is shown in Figure 1.7. The amplitude of the plane wave
exp[i(k 0 x ÿ ù 0 t)] is modulated by the function B(x, t) of equation (1.22),
which has a maximum when (x ÿ v g t) equals zero, i.e., when x ˆ v g t. The
nodes of B(x, t) nearest to the maximum occur when (x ÿ v g t)Äk=2 equals
ð, i.e., when x is (2ð=Äk) from the point of maximum amplitude. If we
consider the half width of the wave packet between these two nodes as a
measure of the uncertainty Äx in the location of the wave packet and the width
(k 2 ÿ k 1 ) of the square pulse A(k) as a measure of the uncertainty Äk in the
value of k, then the product of these two uncertainties is
ÄxÄk ˆ 2ð

Uncertainty relation
We have shown in the two examples above that the uncertainty Äx in the
position of a wave packet is inversely related to the uncertainty Äk in the wave
numbers of the constituent plane waves. This relationship is generally valid and