Page 57 - Electrical Properties of Materials

40 The electron

and this is nothing else but our good old wave solution, at least as a function
of time, if we equate

E = ω. (3.12)
This is actually something we have suggested before [eqn (2.27)] by recourse
to Planck’s formula. So we may call E the energy of the electron. However,
before making such an important decision let us investigate eqn (3.10) which
also contains E. We could rewrite eqn (3.10) in the form:
2
2
– ∇ + V ψ = Eψ. (3.13)
2m
The second term in the bracket is potential energy, so we are at least in
2
good company. The first term contains ∇ , the differential operator you will
have met many times in electrodynamics. Writing it symbolically in the form:

2 2 1 2
– ∇ = (–i ∇) , (3.14)
2m 2m
we can immediately see that by introducing the new notation,

p =–i ∇, (3.15)
and calling it the ‘momentum operator’ we may arrive at an old familiar
relationship:
p 2
= kinetic energy. (3.16)
2m
Thus, on the left-hand side of eqn (3.13) we have the sum of kinetic and
potential energies in operator form and on the right-hand side we have a con-
stant E having the dimensions of energy. Hence, we may, with good conscience,
interpret E as the total energy of the electron.
You might be a little bewildered by these definitions and interpretations, but
you must be patient. You cannot expect to unravel the mysteries of quantum
mechanics at the first attempt. The fundamental difficulty is that first steps in
quantum mechanics are not guided by intuition. You cannot have any intuitive
feelings because the laws of quantum mechanics are not directly experienced
in everyday life. The most satisfactory way, at least for the few who are math-
ematically inclined, is to plunge into the full mathematical treatment and leave
the physical interpretation to a later stage. Unfortunately, this method is lengthy
and far too abstract for an engineer. So the best we can do is to digest alternately
a little physics and a little mathematics and hope that the two will meet.

3.4 The electron as a wave
Let us look at the simplest case when V = 0 and the electron can move only
in one dimension. Then eqn (3.13), which is often called the time-independent
Schrödinger equation, reduces to
2
2
∂ ψ
+ Eψ = 0. (3.17)
2m ∂z 2