Page 61 - Electrical Properties of Materials

44 The electron

3.7 Two analogies
Without the help of Schrödinger’s equation, we could not have guessed
how electrons behaved when meeting a potential barrier. But having found
the solutions in the form of propagating and exponentially decaying waves,
a physical picture, I hope, is emerging. There is always a physical pic-
ture if you are willing to think in terms of waves. Then it is quite nat-
ural that discontinuities cause reflections and only a part of the wave is
transmitted.
The concepts are not appreciably more difficult than those needed to de-
scribe the motion of classical electrons, but you need time to make yourself
familiar with them. ‘Familiarity breeds contempt’ may very well apply to arts
subjects, but in most branches of science the saying should be reformulated as
‘familiarity breeds understanding’ or, more poignantly, as ‘lack of familiarity
breeds bewilderment’.
Assuming that you have already developed some familiarity with waves,
it may help to stress the analogy further. If we went a little more deeply into
the mathematical relationships, we would find that the problem of an electron
meeting a potential barrier is entirely analogous to an electromagnetic wave
meeting a new medium. Recalling the situations depicted in Figs. 1.4 to 1.6,
the analogies are as follows.
1. There are two semi-infinite media (Fig. 1.4); electromagnetic waves
propagate in both of them. Because of the discontinuity, a certain part of the
wave is reflected. This is analogous to the electron meeting a potential barrier
(Fig. 3.2) with an energy E > V 2 . Some electrons are reflected because of the
presence of a discontinuity in potential energy.
2. There are two semi-infinite media (Fig. 1.5); electromagnetic waves may
propagate in the first one but not in the second one. The field intensities are,
however, finite in medium 2 because the electromagnetic wave penetrates to
a certain extent. This is analogous to the electron meeting a potential barrier
(Fig. 3.2) with an energy E < V 2 . In spite of not having sufficient energy, some
electrons may penetrate into region 2.
3. There are two semi-infinite media separated from each other by a third
medium (Fig. 1.6); electromagnetic waves may propagate in media 1 and 3
but not in the middle one. The wave incident from medium 1 declines in
medium 2 but a finite amount arrives and can propagate in medium 3. This
is analogous to the electron meeting a potential barrier shown in Fig. 3.3,
with an energy E < V 2 . In spite of not having sufficient energy some elec-
trons may cross region 2 and may appear and continue their journey in
region 3.
Instead of taking plane waves propagating in infinite media, one might make
the analogy physically more realizable though mathematically less perfect, by
employing hollow metal waveguides. Discontinuities can then be represented
by joining two waveguides of different cross-sections, and the exponentially
decaying wave may be obtained by using a cut-off waveguide (of dimension
smaller than half free-space wavelength). Then all the above phenomena can
be easily demonstrated in the laboratory.