The wavefunction as a measure of our knowledge

In 52.2 we tended to regard the wavefunction as describing a
particular system (in fact, just a single particle). Suppose, however,
we take the view that the wavefunction instead describes our
knowledge of the system. By implication the system might then be
thought of to have other properties of which, at the time, we are
ignorant. For example, the particle may actually be at a specific
position, whereas we know only that it has a certain probability of
being in some region. This, at first sight, appears to be a very
reasonable view. It is indeed the situation that occurs whenever
probability aspects arise in non-quantum situations.
To have a trivial example of this, let us suppose that I am in a
room with 10oO other people. Assume also that I know the lo00 is
made up of 498 French men, 2 French girls, 200 Norwegian men
and 300 Norwegian girls. With this information I would know that
the probability of the person immediately behind me being French
was one in two. Now suppose that I looked at the person behind
me and saw that she was female. The probability of the person
being French would immediately change to one in fifty.
If the situation in quantum theory is of a similar nature then the
issue of the reduction of the wavefunction, raised in 52.3, immediately
goes away. When the wavefunction is just an expression
of our knowledge of the truth, then it is not surprising, and is even
expected, that is should suddenly change to something else when a
measurement is made. A measurement has simply changed our
knowledge (this of course is normally the purpose of making
measurements).
Superficially attractive though this view of the wavefunction may
be, it is in one very important respect inadequate. It cannot explain
the phenomenon of interference. We remind ourselves here that
there is abundant experimental evidence for interference effects and, contrary to what appears to happen in some discussions.of the
interpretation problems of quantum theory, they cannot be ignored.
Wavefunctions which merely represent our knowledge of a system
cannot interfere. We can see this immediately in the case of the
potential barrier experiment. There we require that ‘something’
follows both routes to the detector. That ‘something’ cannot be our
knowledge, which, if it is anywhere, is in our brain. If the particle
really has followed one route then we are back with the problem
as to how its motion can be influenced by the presence of the other
mirror. It is not an answer to this to say that we know about the
other mirror; the behaviour of the particles surely cannot depend
upon the information contained in the brains of particular
individuals.
We can therefore be sure that, if interference actually occurs, this
interpretation of the wavefunction must be wrong. However, the
form of the qualification used here is important. What we know is
that the results of our observation can be predicted from the
calculation of the interference effect. It looks as though interference
is actually happening but it is possible that this is not so,
but that, instead, the calculation just ‘happens’ to give the right
answer. A simple analogy might help here. An umpire at a cricket
match counts the number of balls that have been bowled by placing
pebbles in his pocket, one for each ball. When six pebbles are in
the pocket he calls ‘over’ and play changes ends. Now the reason
for this change is not directly anything to do with pebbles in the
umpire’s pocket, it is because six balls have been bowled and the
rules say that play changes ends every six balls. The pebbles can be
used by the umpire to make the calculation because of the rules of
arithmetic which ensure that the right answer will be obtained. It
could be that a similar thing is happening with the interference
calculation; it gives the right answer but the real reason for the
experimental facts lies elsewhere.
Where? Clearly we must look at the hidden information-at the
properties not contained in our knowledge of the system, and
therefore not in the wavefunction. We are then in the domain of
hidden variable theories which we discuss in detail in Chapter Five.
However, to complete this section we should look ahead and note
that such theories do not in fact eliminate the need for an interfering
wavefunction. Indeed, it is inconceivable that any theory could
successfully reproduce all the correct effects of interference unless the interference actually happens. Thus, although it was important
to mention the reservations of the previous paragraph, I believe
they can now be forgotten.