## Band Structure p.10

On of the themes of these pages is that classical mechanics illuminates
quantum mechanics. Is band structure an exception? In the classical
motion of an electron through a lattice, we find what appears to be a random walk.
Classically the motion of the electron depends so sensitively on the
initial conditions that a formula for the trajectory would be of
little use. Instead we are led to consider statistical
concepts like mean free path.
In quantum mechanics, however, we have Block wavefunctions with
lock-step periodicity that extends throughout the crystal. There seems
to be no scattering of our Block states by the periodic potential...
instead the effect is simply to make *E*(**k**)
a complicated function. Where is the classical analogy?

To gain some appreciation for what is happening, consider the
motion of visible light through, say, a diamond crystal.
Certainly the light is scattered by each electron and nucleus
that makes up the lattice, but we see straightforward
behavior. The effect of all of this scattering is present
but seems to produce only a minor change:
a non-unit the index of refraction. Of course, if we use light
with shorter wavelength (say X-rays) we begin to see Bragg scattering
from the lattice. Similarly for electron waves: for small **k**
where the wavelength is large compared to the lattice spacing,
we see straightforward behavior (that nevertheless is the result
of multiple scattering). However, when the wavelength
of the light is comparable to the size of the lattice (e.g.,
near the first zone boundary) we see major effects: band gaps=non-propagation
of waves that satisfy the Bragg condition.

Of course, Ehrenfest's theorem is still correct: if we localize
a probability blob to say, 1/10 of a lattice spacing (which will
require highly excited electron states), that blob of probability will follow
the classical trajectory. The indefinite initial position and velocity
(as required by Heisenberg's uncertainty)
means varying initial conditions. Classically such different initial
conditions result in divergent trajectories. Expect the probability
blob to "quickly" spread out.

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