Wave Mechanics 199
it is then possible to specify the exact position and momentum of the
body for all future times as long as one knows the forces acting on the
body. This programme is no longer possible within the framework of
quantum mechanics even is one knows all the forces on the body. The
uncertainty principle only permits a partial or approximate knowledge
of the location and momentum of the body. For instance, one might
know that the body is between x and x + ∆x with a momentum between
p and p + ∆p where ∆p ∆x = h. It is now impossible to specify exactly
where the particle will be some time later. A particle at x with
momentum p will behave differently than a particle at x + ∆x with
momentum p + ∆p. The uncertainty will tend to increase with time.
The uncertainty principle forces a change in our description. Instead
of specifying the exact location and momentum of the particle as a
function of the time as was done in classical physics, we are now forced
to describe the particle in terms of the probability of finding it at some
place, x, with a momentum, p. Within the framework of quantum
mechanics, the probability of determining the particle’s position and
momentum is described quantum mechanically by the probability
amplitude or wave function, ψ(x,y,z), which has a unique value for each
position in space x, y and z.
The equations describing the behaviour of the wave function, ψ, is
the Schrödinger equation referred to earlier. When Schrödinger first
discovered his equations he did not connect the wave function, ψ(x, y, z)
with probability but rather interpreted it as the density of the electron
cloud, which he considered to be spread out through space. It was Max
Born who pointed out that the correct interpretation was to continue to
assume that the electron is a point particle and to regard ψ as a measure
of the probability of detecting the point electron at some point in space.
He showed, in fact, that the probability of finding the electron located
at the point in space x, y and z is just the absolute value of the
wave function multiplied by itself, |ψ (x, y, z)|^2. He also showed that
Schrödinger’s ψ determines the probability that the particle has a
particular value of the momentum can also be determined from ψ but
involves a more complicated mathematical operation than multiplying
ψ by itself. These mathematical details need not concern us. The
important point is that once one knows a particle’s wave function,
ψ(x, y, z), one can determine the probability it will have a particular
momentum and a particular position.
The wave behaviour of the electron is due to the fact that its
probability amplitude, ψ, behaves like a wave. The Schrödinger equation