Page 24 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.3 Dispersion of a wave packet 15

the previous analysis, if we select a speci®c form for the modulating function
G(ù) such as a gaussian or a square pulse distribution, we can show that the
product of the uncertainty Ät in the time variable and the uncertainty Äù in
the angular frequency of the wave packet has a lower bound of order unity, i.e.
ÄtÄù > 1 (1:25)
This uncertainty relation is also a property of Fourier transforms and is valid
for all wave packets.

1.3 Dispersion of a wave packet
In this section we investigate the change in contour of a wave packet as it
propagates with time.
The general expression for a wave packet Ø(x, t) is given by equation
(1.11). The weighting factor A(k) in (1.11) is the inverse Fourier transform of
Ø(x, t) and is given by (1.12). Since the function A(k) is independent of time,
we may set t equal to any arbitrary value in the integral on the right-hand side
of equation (1.12). If we let t equal zero in (1.12), then that equation becomes

1 1 ÿikî
A(k) ˆ p  Ø(î, 0)e dî (1:26)
2ð ÿ1
where we have also replaced the dummy variable of integration by î. Substitu-
tion of equation (1.26) into (1.11) yields
1

1 i[k(xÿî)ÿùt]
Ø(x, t) ˆ Ø(î, 0)e dk dî (1:27)
2ð
ÿ1
Since the limits of integration do not depend on the variables î and k, the order
of integration over these variables may be interchanged.
Equation (1.27) relates the wave packet Ø(x, t) at time t to the wave packet
Ø(x, 0) at time t ˆ 0. However, the angular frequency ù(k) is dependent on k
and the functional form must be known before we can evaluate the integral
over k.
If ù(k) is proportional to jkj as expressed in equation (1.6), then (1.27) gives

1

1
Ø(x, t) ˆ Ø(î, 0)e ik(xÿctÿî) dk dî
2ð
ÿ1
The integral over k may be expressed in terms of the Dirac delta function
through equation (C.6) in Appendix C, so that we have