Page 67 - Electrical Properties of Materials

50 The electron
~
t = h/2.84 eV = 1.5 × 10 –15 s. Now, you may ask, what can an electron do
in a time interval as short as 1.5 × 10 –15 s? Quite a lot; it can get comfortably
from one atom to the next. And remember all this on borrowed energy. So if
there was a barrier of (say) five electron volts, our electron could easily move
over it, and having scaled the barrier it could return the borrowed energy, and
no one would be able to find out how the electron made the journey. Thus,
from a purely philosophical argument we could make up an alternative picture
of tunnelling. We may say that tunnelling across potential barriers comes about
because the electron can borrow energy for a limited time. Is this a correct de-
scription of what happens? I do not know, but it is a possible description. Is
it useful? I suppose it is always useful to have various ways of describing the
same event; that always improves understanding. But the crucial test is whether
this way of thinking will help in arriving at new conclusions which can be ex-
perimentally tested. For an engineer the criterion is even clearer; if an engineer
can think up a new device, using these sorts of arguments (e.g. violating en-
ergy conservation for a limited time) and the device works (or even better it
can be sold for ready money) then the method is vindicated. The end justifies
the means, as Machiavelli said.
Theoretical physicists, I believe, do use these methods. In a purely particle
description of Nature, for example, the Coulomb force between two electrons
is attributed to the following cause. One of the electrons borrows some energy
to create a photon that goes dutifully to the other electron, where it is absorbed,
returning thereby the energy borrowed. The farther the two electrons are from
each other, the lesser the energy that can be borrowed, and therefore lower
frequency photons are emitted and absorbed. Carrying on these arguments (if
you are interested in more details ask a theoretical physicist) they do manage
to get correctly the forces between electrons. So there are already some people
who find it useful to play around with these concepts.
This is about as much as I want to say about the philosophical role of the
electron. There are, incidentally, a number of other points where philosophy
and quantum mechanics meet (e.g. the assertion of quantum mechanics that no
event can be predicted with certainty, merely with a certain probability), but
I think we may have already gone beyond what is absolutely necessary for the
education of an engineer.

Exercises
3.1. An electron, confined by a rigid one-dimensional poten- eqn (3.42). Calculate the average values of
tial well (Fig. 3.4 with V 1 = ∞) may be anywhere within z, (z – z ) , p, ( p – p ) .
2
2
the interval 2a. So the uncertainty in its position is x =2a.
There must be a corresponding uncertainty in the momentum [Hint: Use eqn (3.43). The momentum operator in this one-
dimensional case is –i ∂/∂z.]
of the electron and hence it must have a certain kinetic en-
ergy. Calculate this energy from the uncertainty relationship 3.3. The classical equivalent of the potential well is a particle
and compare it with the value obtained from eqn (3.42) for bouncing between two perfectly elastic walls with uniform
the ground state. velocity.
3.2. The wave function for a rigid potential well is given by (i) Calculate the classical average values of the quantities
eqns (3.31) and (3.32) and the permissible values of k by enumerated in the previous example.