Page 14 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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1.1 Wave motion 5
For dispersive wave motion, the angular frequency ù(k) is not proportional
to |k|, so that the phase velocity v ph varies from one component plane wave to
another. Since the phase velocity in this situation depends on k, the shape of
the composite wave changes with time. An example of dispersive wave motion
is a beam of light of mixed frequencies traveling in a dense medium such as
glass. Because the phase velocity of each monochromatic plane wave depends
on its wavelength, the beam of light is dispersed, or separated onto its
component waves, when passed through a glass prism. The wave on the surface
of water caused by dropping a stone into the water is another example of
dispersive wave motion.
Addition of two plane waves
As a speci®c and yet simple example of composite-wave construction and
behavior, we now consider in detail the properties of the composite wave
Ø(x, t) obtained by the addition or superposition of the two plane waves
exp[i(k 1 x ÿ ù 1 t)] and exp[i(k 2 x ÿ ù 2 t)]
Ø(x, t) e i(k 1 xÿù 1 t) e i(k 2 xÿù 2 t) (1:7)
We de®ne the average values k and ù and the differences Äk and Äù for the
two plane waves in equation (1.7) by the relations
k 1 k 2 ù 1 ù 2
k ù
2 2
Äk k 1 ÿ k 2 Äù ù 1 ÿ ù 2
so that
Äk Äk
k 1 k , k 2 k ÿ
2 2
Äù Äù
ù 1 ù , ù 2 ù ÿ
2 2
Using equation (A.32) from Appendix A, we may now write equation (1.7) in
the form
Ø(x, t) e i(kxÿùt) [e i(ÄkxÿÄùt)=2 e ÿi(ÄkxÿÄùt)=2 ]
Äkx ÿ Äùt
2 cos e i(kxÿùt) (1:8)
2
Equation (1.8) represents a plane wave exp[i(kx ÿ ùt)] with wave number k,
angular frequency ù, and phase velocity ù=k, but with its amplitude modulated
by the function 2 cos[(Äkx ÿ Äùt)=2]. The real part of the wave (1.8) at some
®xed time t 0 is shown in Figure 1.2(a). The solid curve is the plane wave with
wavelength ë 2ð=k and the dashed curve shows the pro®le of the amplitude
of the plane wave. The pro®le is also a harmonic wave with wavelength
