Page 18 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.2 Wave packet 9

two speci®c examples for the functional form of A(k). However, in order to
evaluate the integral over k in equation (1.11), we also need to know the
dependence of the angular frequency ù on the wave number k.
In general, the angular frequency ù(k) is a function of k, so that the angular
frequencies in the composite wave Ø(x, t), as well as the wave numbers, vary
from one plane wave to another. If ù(k) is a slowly varying function of k and
the values of k are con®ned to a small range Äk, then ù(k) may be expanded
in a Taylor series in k about some point k 0 within the interval Äk

dù 1 d ù
2
2
ù(k) ˆ ù 0 ‡ (k ÿ k 0 ) ‡ (k ÿ k 0 ) ‡ (1:13)
dk 0 2 dk 2 0
where ù 0 is the value of ù(k)at k 0 and the derivatives are also evaluated at k 0 .
We may neglect the quadratic and higher-order terms in the Taylor expansion
(1.13) because the interval Äk and, consequently, k ÿ k 0 are small. Substitu-
tion of equation (1.13) into the phase for each plane wave in (1.11) then gives

dù
kx ÿ ùt (k ÿ k 0 ‡ k 0 )x ÿ ù 0 t ÿ (k ÿ k 0 )t
dk 0
" #

dù
ˆ k 0 x ÿ ù 0 t ‡ x ÿ t (k ÿ k 0 )
dk 0
so that equation (1.11) becomes
Ø(x, t) ˆ B(x, t)e i(k 0 xÿù 0 t) (1:14)
where

1 1 i[xÿ(dù=dk) 0 t](kÿk 0 )
B(x, t) ˆ p  A(k)e dk (1:15)
2ð ÿ1
Thus, the wave packet Ø(x, t) represents a plane wave of wave number k 0 and
angular frequency ù 0 with its amplitude modulated by the factor B(x, t). This
modulating function B(x, t) depends on x and t through the relationship
[x ÿ (dù=dk) 0 t]. This situation is analogous to the case of two plane waves as
expressed in equations (1.7) and (1.8). The modulating function B(x, t)moves
in the positive x-direction with group velocity v g given by

dù
v g ˆ (1:16)
dk 0
In contrast to the group velocity for the two-wave case, as expressed in
equation (1.9), the group velocity in (1.16) for the wave packet is not uniquely
de®ned. The point k 0 is chosen arbitrarily and, therefore, the value at k 0 of the
derivative dù=dk varies according to that choice. However, the range of k is