Interference

We shall next consider the quantum theoretical description of the second type of barrier experiment discussed in Chapter One. In
this, we recall, there were mirrors which could bring both the
reflected and the transmitted particles to the same set of detectors.
We begin then with the same initial state as before (figure ll(a))
and follow the wavefunction to the situation shown in figure 1 l ( d ) .
Here, to a good approximation, the wavefunction can be regarded
as a sum of two wavefunctions, one giving the left-hand peak and
the other the right-hand peak. Note that the operation of adding
the two wavefunctions is rather trivial at this stage since, at any
given point of space, at most one of the two wavefunctions which
are added is different from zero. In the subsequent motion each of
the two peaks will change independently; in fact they will move in
a manner closely resembling the classical motion of a free particle.
(It is irrelevant here that the area under each peak is not actually
equal to one.)
Eventually, if the mirrors are present, the peaks will come
together in the neighbourhood of the detectors. At this stage the
addition is no longer trivial since both wavefunctions are different
from zero at the same place. This means that the feature mentioned
at the end of 52.2 becomes relevant, and the probability resulting
from the two wavefunctions is not equal to the sum of the probabilities
associated with the separate wavefunctions.
We have here an example of an extremely important
phenomenon known as ‘interference’. It occurs in a wide range of
physical situations even where quantum effects are not relevant. As
an example, we can think of two pebbles being dropped onto the
surface of a still pond. Ripples will spread out from the points of
impact. At some positions on the pond the ‘ups’ and the ‘downs’
from the two circular wave patterns will always come at the same
time and the wave will therefore be enhanced. At others they will
be ‘out of phase’, i.e. an ‘up’ from one will arrive at the same time
as a ‘down’ from the other, in which case they will cancel each
other and the water will remain still. Figure 12 illustrates this
situation.
In our quantum mechanics problem the situation is rather more
complicated since we are not just adding numbers, which can be
positive or negative, but adding ‘lines’, and we recall that the result
depends on the angle between the lines. On the other hand, if we
think just of the real parts of the wavefunctions, then what happens
is very similar to the case of water waves, The precise forms of the two wavefunctions to be added will depend on the length of the
path to any particular detector (see figure 4, for example). It
follows that the nature of the interference observed will depend on
which detector is considered. Certainly, in general, the probability
resulting from the sum of the two wavefunctions will be different
from the sum of the probabilities coming from each separately. This is in accordance with the observations which we found so
surprising in 41.4.
Detailed calculations yielding precise results are, of course,
possible. Similar calculations can be done for other situations in
which quantum mechanical interference occurs, and where the
results can be verified by experiments. Of particular importance are
experiments where electrons are scattered off crystals. Here the
interference is between parts of the wavefunction scattered off
different sites in the crystal. Comparison of the results with
calculated predictions reveals information on the structure of the
crystal.
A brief historical note is of interest here. The long-standing
conflict between a corpuscular theory of light (favoured by Isaac
Newton) and a wave theory was generally believed to have been
settled in favour of the latter by observation of interference effects
when light was passed through two slits (see figure 13). Interference
implied waves. It was therefore a shock when electrons, long
established as particles, were also found to show interference
effects. This schizophrenic behaviour became known as ‘particlewave
duality’. The same duality applies to electromagnetic
radiation, of which light is an example. The ‘particles’ of light are
called photons. In our potential barrier example, the particle nature
is seen most naturally in the first set of experiments where the
particle is observed either to be transmitted or reflected. The wave
nature is seen in the second set, where there is evidence for
interference effects.
Quantum theory successfully incorporates both features and
enables us to calculate correctly all microscopic phenomena that do
not involve ‘relativistic’ effects. A brief review of some of the
successes of the theory is given in the next section, with which we
conclude this chapter.