The wavefunction as part of external reality

We now want to consider the possibility that the wavefunction
should be treated rather more seriously than in the preceding two
sections, so that we can use it to tell us something about the
external reality. We shall try to regard the wavefunction not as just
a description of a statistical ensemble, as in 53.1, or as a catalogue
of our information about a system, as in 53.2, but as something
that really exists, something that is, indeed, part of the external
reality which we observe.
There are at least three good reasons why we should want to consider
this assumption. First, since the classical picture of a single
particle, always having a precise position and following a specific
path, is not compatible with the observations described in 01.4, we
do not have any other object available for our representation of
reality. Secondly, the evidence that wavefunctions can interfere
strongly suggests that they are real, e.g. just like ripples on the surface
of a pond. In order to understand the third reason we need to
know about certain symmetry properties that have to be imposed
on wavefunctions describing more than one particle. If we have two
identical particles, e.g. two electrons, then in classical mechanics
we could distinguish them, for example, by their positions. In
quantum theory, on the other hand, they are described by a
wavefunction which tells us the probability of finding an electron
at one place and an electron at another place; in no way are the two
electrons distinguished. This means that the wavefunction must be
symmetrical in the two electrons, i.e. it must not change if we interchange
them. Actually, the truth is a little different from this
because in some particular cases the wavefunction has to change its
sign. Such a change, however, does not alter any of the physics,
which is determined by the square of the magnitude of the
wavefunction. A more detailed discussion of this is given in
Appendix 4. Here we merely note that the symmetry properties give rise to important, testable, predictions, which have been verified
and which would be very hard to understand without the assumption
that wavefunctions have a real existence.
Our tentative picture of the potential barrier experiment is
therefore that of a wavefunction which has a value that varies with
the point of space being considered. We are familiar with quantities
of this type, e.g. the temperature of the air at different points of
a room, or the number of flies per unit volume in a field of cattle.
Actually the wavefunction is a little different since, as we recall, it
is a line or, alternatively, two numbers at each point of space. This
fact, however, does not affect the present discussion, so we shall
continue to refer simply to the value of the wavefunction.
As is illustrated for example in figure 11, the wavefunction'is in
general not constant but changes with time. Again this is a concept
with which we are familiar; the temperatures at various points in
a room, for example, will similarly change with time, e.g. when the
heating has been switched off. We therefore have a simple picture
of reality, with the wavefunction describing something that actually
happens.
There are, however, two difficulties associated with this picture.
The first of these is due to the fact that the world does not consist
of just one particle. We remember that the wavefunction we have
used so far was specifically designed to treat only one particle. How
do we generalise this to accommodate additional particles?
Consider a world of two particles, which we shall call A and B.
As a first guess we might try having a wavefunction for particle A
and a separate and independent one for particle B. Then the
probability of finding A at some point would not depend on the
position of B. This is reasonable for particles that are genuinely
independent, i.e. not interacting. It is, however, quite unreasonable,
and is indeed false, for particles that are interacting. In this
case the wavefunction must depend on rwo positions. It will then
tell us the probability for finding particle A at one position and partide
B at the other. (Some further details are given in Appendix 4.)
One can express this by saying that the wavefunction does not exist
in the usual space of three dimensions but in a space of two-timesthree
dimensions. It is no longer true to say that at a particular
point of space the wavefunction has a particular value. Rather we
have to say that, associated with every two points of space (or, if we prefer to express it this way, with every point of a sixdimensional
space) there is a particular value for the wavefunction.
Of course, we cannot stop at two particles and must go on to
include 3,4, etc, with the wavefunction depending on the corresponding
number of points, 9, 12, etc, in space. At this stage the
wavefunction starts to look more like a mathematical device than
something that is part of the real world. Certainly it is not now of
the form of the familiar quantities mentioned earlier. These are
local, i.e. at a single point of space there is a number which is the
temperature. The wavefunction, on the contrary, is non-local; in
order to establish its value we need to give many positions in space.
We shall find this non-locality occurring again in our discussion.
It should be noted here that the two-particle wavefunction is
not, in general, simply a product of two one-particle wavefunctions.
To understand this distinction we recall that the square of the
magnitude of the wavefunction gives the probability of finding
a particle at each of the two points. If the particles are quite
independent, and not in any way correlated in position, then the
probability of finding a particle at a point P will not depend on the
position of the other. In such a case the wavefunction will be a
simple product of two wavefunctions, each depending upon one
position. In most real situations, however, particles interact and
therefore their positions are correlated. The wavefunction is then
not of the product type but is, rather, one function with an explicit
dependence upon two positions. Again we refer to Appendix 4 for
further details.
The second difficulty that arises when we regard the wavefunction
as part of reality is one to which we have already referred,
the process of reduction of the wavefunction. As we saw in $2.3,
the wavefunction changes when a measurement is made. This
change appears to be sudden and discontinuous. It is also very nonlocal
in the sense that measurements at one point of space can
change the wavefunction at other points, regardless of how far
away these might be. The measurement by means of a detector on
the right-hand side of the potential barrier provides a good example
of this. If this flashes it means that the particle has been reflected,
so the piece of the wavefunction on the left (e.g. in figure 11) immediately
becomes zero. This, at least, appears to be what is happening.
Whenever a measurement is made on a system described by a wavefunction, then one of the possible values consistent with the
probability distribution is obtained. The measurement somehow
selects part of the wavefunction. We cannot be content, however,
with merely postulating that this happens. We must ask how it
happens. In particular, we have claimed that quantum mechanics
is a universal theory and applies to everything. It should therefore
apply to the apparatus which we use to make a ‘measurement’, and
should, therefore, contain the answer to our question-that is,
quantum mechanics should be able to explain how the wavefunction
reduces. In fact, however, it says very clearly that the
wavefunction cannot reduce! Such a startling fact deserves another
section.