Monday, July 2, 2007

The potential barrier and the breakdown of determinism

We now want to describe a set of simple experiments which
demonstrate the crucial features of quantum phenomena. To begin
we suppose that we have a flat table on which there is a smooth ‘hill’, This is illustrated in figure 1. If we roll a small ball, from the
right, towards the hill then, for low initial velocities, the ball will
roll up the hill, slowing down as it does so, until it stops and then
rolls back down again. In this case we say that the ball has been
reflected. For larger velocities, however, the ball will go right over
the hill and will roll down the other side; it will have been
transmitted.

Now we introduce quantum physics. The simple result expressed
by equation (l.l), which we obtained from experiment and which
is in agreement with the laws of classical mechanics, is not in fact
correct. For example, even when v < Vthere is a possiblity that the
particle will pass through the barrier. This phenomenon is sometimes
referred to as quantum tunnelling. The reason why we
would not see it in our simple laboratory experiment is that with
objects of normal sizes (which we shall refer to as ‘macroscopic’
objects), i.e. things we can hold and see, the effect is far too small
to be noticed. Whenever v is measurably smaller than V the
probability of transmission is so small that we can effectively say
it will never happen.

With ‘microscopic’ objects, i.e. those with atomic sizes and
smaller, the situation is very different and equation (1.1) does not
describe the results except for sufficiently small, or sufficiently
large, velocities. For velocities close to V we find, to our surprise,
that the value of v does not tell us whether or not the particle will
be transmitted. If we repeat the experiment several times, always
with a fixed initial velocity (v) we would find that in some cases the
particle is reflected and in some it is transmitted. The value of v
would no longer determine precisely the fate of the particle when
it hits the barrier; rather it would tell us the probability of a particle
of that velocity passing through. For low velocities the probability
would be close to zero, and we would effectively be in the classical
situation; as the velocity rose towards V the probability of
transmission would rise steadily, eventually becoming very close to
unity for v much larger than V, thus again giving the classical
result.

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