f/Three possible standing-
wave patterns for a particle in a
box.small, we can approximate this expression as a derivative,vg=
df
dz.
This expression is usually taken as the definition of the group veloc-
ity for wave patterns that consist of a superposition of sine waves
having a narrow range of frequencies and wavelengths. In quan-
tum mechanics, withf=E/handz=p/h, we havevg= dE/dp.
In the case of a nonrelativistic electron the relationship between
energy and momentum isE = p^2 / 2 m, so the group velocity is
dE/dp=p/m=v, exactly what it should be. It is only the phase
velocity that differs by a factor of two from what we would have
expected, but the phase velocity is not the physically important
thing.13.3.3 Bound states
Electrons are at their most interesting when they’re in atoms,
that is, when they are bound within a small region of space. We
can understand a great deal about atoms and molecules based on
simple arguments about such bound states, without going into any
of the realistic details of atom. The simplest model of a bound state
is known as the particle in a box: like a ball on a pool table, the
electron feels zero force while in the interior, but when it reaches
an edge it encounters a wall that pushes back inward on it with
a large force. In particle language, we would describe the electron
as bouncing off of the wall, but this incorrectly assumes that the
electron has a certain path through space. It is more correct to
describe the electron as a wave that undergoes 100% reflection at
the boundaries of the box.
Like generations of physics students before me, I rolled my eyes
when initially introduced to the unrealistic idea of putting a particle
in a box. It seemed completely impractical, an artificial textbook
invention. Today, however, it has become routine to study elec-
trons in rectangular boxes in actual laboratory experiments. The
“box” is actually just an empty cavity within a solid piece of silicon,
amounting in volume to a few hundred atoms. The methods for
creating these electron-in-a-box setups (known as “quantum dots”)
were a by-product of the development of technologies for fabricating
computer chips.
For simplicity let’s imagine a one-dimensional electron in a box,
i.e., we assume that the electron is only free to move along a line.
The resulting standing wave patterns, of which the first three are
shown in the figure, are just like some of the patterns we encoun-
tered with sound waves in musical instruments. The wave patterns
must be zero at the ends of the box, because we are assuming the
walls are impenetrable, and there should therefore be zero proba-
bility of finding the electron outside the box. Each wave pattern is
labeled according ton, the number of peaks and valleys it has. In
Section 13.3 Matter as a wave 897