Page 65 - Electrical Properties of Materials

48 The electron

Can we now make a more precise statement about the uncertainty relation-
ship? We could, if we introduced a few more concepts. If you are interested
in the details, you can consult any textbook on quantum mechanics. Here,
I shall merely outline one of the possible ways of deriving the uncertainty
relationship.
First, the average value of a physically measurable quantity, called an
observable, is defined in quantum mechanics as

∗
ψ Aψ d(volume)
V
A = 2 , (3.43)
|ψ| d(volume)
V
where the integration is over the volume of interest, wherever ψ is defined. A is
in general an operator; it is –i ∇ for the momentum and simply r, the radius
vector, for the position. Assuming that Schrödinger’s equation is solved for a
particular case, we know the wave function ψ, and hence, with the aid of eqn
(3.43), we can work out the average and r.m.s. values of both the electron’s
position and of its momentum. Identifying z and p with
2
2
{ (z – z ) } 1/2 and { ( p – p ) } 1/2
respectively, we get

z p h. (3.44)
There is actually another often-used form of the uncertainty relationship

E t h, (3.45)

which may be derived from relativistic quantum theory (where time is on equal
footing with the spatial coordinates) and interpreted in the following way. As-
sume that an electron sits in a higher energy state of a system, for example in
a potential well. It may fall to the lowest energy state by emitting a photon of
energy ω. So if we know the energy of the lowest state, we could work out
the energy of that particular higher state by measuring the frequency of the
The emitted radiation is not mono- emitted photon. But if the electron spends only a time t in the higher state,
chromatic; it covers a finite range then the energy of the state can be determined with an accuracy not greater
of frequencies. than E = h/ t. This is borne out by the measurements.

3.11 Philosophical implications

The advent of quantum mechanics brought problems to the physicist which
previously belonged to the sacred domain of philosophy. The engineer can still
afford to ignore the philosophical implications but by a narrow margin only.
In another decade or two philosophical considerations might be relevant in the
discussion of devices, so I will try to give you a foretaste of the things which
might come.
To illustrate the sort of questions philosophers are asking, take the following
one: We see a tree in the quad, so the tree must be there. We have the evidence
of our senses (the eye in this particular case) that the tree exists. But what
happens when we don’t look at the tree, when no one looks at the tree at all;
does the tree still exist? It is a good question.