Page 55 - Electrical Properties of Materials

38 The electron

of energy levels in strong transverse fields. He produced four papers in quick
succession and noted at the end with quiet optimism:
‘I hope and believe that the above attempt will turn out to be useful for ex-
plaining the magnetic properties of atoms and molecules, and also the electric
current in the solid state.’
∗
∗ In the above discussion the role Schrödinger was right. His equation turned out to be useful indeed. He was
of Schrödinger in setting up modern not exactly right, though. In order to explain all the properties of the solid state
quantum physics was very much exag-
gerated. There were a number of others (including magnetism) two further requisites are needed: Pauli’s principle and
who made comparable contributions, but ‘spin’. Fortunately, both of them can be stated in simple terms, so if we make
since this is not a course in the history ourselves familiar with Schrödinger’s equation, the rest is relatively easy.
of science, and the Schrödinger formula-
tion is adequate for our purpose, we shall
not discuss these contributions. 3.2 Schrödinger’s equation
After such a lengthy introduction, let us have now the celebrated equation
itself. In the usual notation,

m is the mass of the electron, and 2 2 ∂
– ∇ + V =i . (3.4)
V is the potential in which the 2m ∂t
electron moves.
We have a partial differential equation in . But what is ? It is called the
wave function, and
2
| (x, y, z; t)| dx dy dz (3.5)
This interpretation was proposed gives the probability that the electron can be found at time, t,inthe volume
by Max Born, Nobel Prize, 1954. element, dx dy dz, in the immediate vicinity of the point, x, y, z. To show the
2
significance of this function better | | is plotted in Fig. 3.1 for a hypothetical
2
case where | | is independent of time and varies only in one dimension. If
we make many measurements on this system, we shall find that the electron is
always between z 0 and z 4 (the probability of being outside this region is zero),
that it is most likely to be found in the interval dz around z 3 , and it is three
times as probable to find the electron at z 2 than at z 1 . Since the electron must be
somewhere, the probability of finding it between z 0 and z 4 must be unity, that is,
z 4

2
| (z)| dz = 1. (3.6)
z 0
The above example does not claim to represent any physical situation. It is
2
shown only to illustrate the meaning of | | .

0.5

0.4
|Ψ(z)| 2 0.3

Fig. 3.1 0.2
Introducing the concept of the wave
2
function. |ψ(z)| dz proportional to the 0.1
probability that the electron may be 0.0
found in the interval dz at the point z. z 0 z 1 z 2 z 3 z 4 z