c/Part of an infinite sine wave.d/A finite-length sine wave.e/A beat pattern created by
superimposing two sine waves
with slightly different wave-
lengths.2!” The issue that’s involved is a real one, albeit one that could be
glossed over (and is, in most textbooks) without raising any alarms
in the mind of the average student. The present optional section
addresses this point; it is intended for the student who wishes to
delve a little deeper.
Here’s how the now-legendary student was presumably reason-
ing. We start with the equationv=fλ, which is valid for any sine
wave, whether it’s quantum or classical. Let’s assume we already
knowE= hf, and are trying to derive the relationship between
wavelength and momentum:
λ=
v
f=vh
E
=
vh
1
2 mv
2=
2 h
mv
=
2 h
p.
The reasoning seems valid, but the result does contradict the
accepted one, which is after all solidly based on experiment.
The mistaken assumption is that we can figure everything out in
terms of pure sine waves. Mathematically, the only wave that has
a perfectly well defined wavelength and frequency is a sine wave,
and not just any sine wave but an infinitely long sine wave, c. The
unphysical thing about such a wave is that it has no leading or
trailing edge, so it can never be said to enter or leave any particular
region of space. Our derivation made use of the velocity,v, and if
velocity is to be a meaningful concept, it must tell us how quickly
stuff (mass, energy, momentum,... ) is transported from one region
of space to another. Since an infinitely long sine wave doesn’t remove
any stuff from one region and take it to another, the “velocity of its
stuff” is not a well defined concept.
Of course the individual wave peaks do travel through space, and
one might think that it would make sense to associate their speed
with the “speed of stuff,” but as we will see, the two velocities are
in general unequal when a wave’s velocity depends on wavelength.
Such a wave is called adispersivewave, because a wave pulse consist-
ing of a superposition of waves of different wavelengths will separate
(disperse) into its separate wavelengths as the waves move through
space at different speeds. Nearly all the waves we have encountered
have been nondispersive. For instance, sound waves and light waves
(in a vacuum) have speeds independent of wavelength. A water wave
is one good example of a dispersive wave. Long-wavelength water
Section 13.3 Matter as a wave 895