Page 13 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

4 The wave function

The moving harmonic wave ø(x, t) in equation (1.3) is also known as a
plane wave. The quantity (kx ÿ ùt) is called the phase. The velocity ù=k is
known as the phase velocity and henceforth is designated by v ph , so that
ù
v ph ˆ (1:4)
k

Composite wave
A composite wave is obtained by the addition or superposition of any number
of plane waves
n
X
Ø(x, t) ˆ A j e i(k j xÿù j t) (1:5)
jˆ1
where A j are constants. Equation (1.5) is a Fourier series representation of
Ø(x, t). Fourier series are discussed in Appendix B. The composite wave
Ø(x, t) is not a moving harmonic wave, but rather a superposition of n plane
waves with different wavelengths and frequencies and with different ampli-
tudes A j . Each plane wave travels with its own phase velocity v ph, j , such that
ù j
v ph, j ˆ
k j
As a consequence, the pro®le of this composite wave changes with time. The
wave numbers k j may be positive or negative, but we will restrict the angular
frequencies ù j to positive values. A plane wave with a negative value of k has
a negative value for its phase velocity and corresponds to a harmonic wave
moving in the negative x-direction. In general, the angular frequency ù
depends on the wave number k. The dependence of ù(k) is known as the law
of dispersion for the composite wave.
In the special case where the ratio ù(k)=k is the same for each of the
component plane waves, so that
ù 1 ù 2 ù n
ˆ ˆ ˆ
k 1 k 2 k n
then each plane wave moves with the same velocity. Thus, the pro®le of the
composite wave does not change with time even though the angular frequencies
and the wave numbers differ. For this undispersed wave motion, the angular

frequency ù(k) is proportional to jkj
ù(k) ˆ cjkj (1:6)
where c is a constant and, according to equation (1.4), is the phase velocity of
each plane wave in the composite wave. Examples of undispersed wave motion
are a beam of light of mixed frequencies traveling in a vacuum and the
undamped vibrations of a stretched string.