Can quantum mechanics be changed so that it will reduce wavefunctions?

In the quantum theoretical decription of the potential barrier
experiment, which we discussed in @2.3 and 2.4, the wavefunction
split into two pieces, one travelling to the left and one to the right.
This behaviour was good because both pieces were needed to
explain the interference effects. However, it is possible to modify
the Schrodinger equation so that, after a certain time, the form of
the wavefunction changes: one of the peaks grows and the other
falls to zero. If, for example, it is the right-hand peak that remains,
then the equation will have predicted that the particle is reflected.
In this way, reduction of the wavefunction becomes a consequence
of the modified equation. To obtain the probabilistic element which
is vital for agreement with observation it is necessary that the
modification contains some randomly chosen contribution. Then it
is possible to arrange things so that either one of the peaks remains,
with a probability proportional to its original area. In this way we
obtain complete agreement with observation, and we have
automatic reduction of the wavefunction.
Actually, what we have described here for one particular
example can be done in general. Suitable additional terms can be
added to the Schrodinger equation, so that the wavefunction
automatically reduces to the form associated with a particular value
for some measured quantity, always with the correct probability
distribution. These extra terms must contain a random input. There
is also a free constant which can be used to fix the overall
magnitude of the new effects; this determines how long it takes for
the reduction to occur. Some further details of the very pretty
mathematics involved are given in Appendix 7.
At first sight all this appears to be just what we require for a
theory of wavefunction reduction. On closer examination, however,
it is clearly seen to be very unsatisfactory. The first reason for this concerns the time scale which is required for the reduction to
occur. As noted above, this can be adjusted to any desired value
by suitable choice of the magnitude of the extra terms in the
equation. However, no choice can satisfy the experimental constraints,
because these are mutually contradictory. On the one
hand, it is sometimes observed that reduction takes place very
rapidly, whereas, on the other hand, the observation of interference
effects from radio waves that have travelled distances of the
order of the size of the galaxy requires that the reduction time must
be very long. No time scale for automatic reduction of the
wavefunction is compatible with all observations.
The second reason why these ideas are unsatisfactory is that the
wavefunction has to reduce to a form appropriate for any type of
measurement. Hence the particular terms that have to be put into
the Schrodinger equation depend upon what is going to be
measured. In our example we have always thought in terms of
position measurements, but we could instead decide to measure
velocities. This would require a very different type of wavefunction
reduction.
It is worth introducing here another type of experiment, totally
different from anything we have met before, which illustrates this
last point very well and which will also be of use later. Many
particles have a ‘spin’, which always has a constant magnitude. For
example, we shall consider electrons, where the magnitude of the
spin, measured in suitable units, is always 1/2. (Appendix 8 gives
some further details.) The only variable associated with the spin is
its direction. (It is convenient to think of this as the direction of the
axis of a spinning top.) In order to ascertain this direction we
measure the spin along any line in space. It is a consequence of
quantum theory that, in such a measurement, we will always find
one of two values, + 1/2, corresponding to the spin being along the
chosen line, and - 1/2, corresponding to its being in the opposite
direction (see figure 17). Thus, when we make a measurement, the
wavefunction will reduce to the form corresponding to plus or
minus 112 along the line chosen. As we have stated above, it is
possible for this wavefunction reduction to happen automatically if
quantum theory is suitably modified. However, the final form of
the reduced wavefunction, and therefore the modification required,
will depend upon which particular line in space is chosen for the
measurement. There cannot be one equation which describes the future evolution of the electron wavefunction, regardless of what
we choose to measure,

It is clear that both these objections to the type of theory
involving automatic reduction of the wavefunction can be met if
the modifications to the standard quantum theory ‘know about’
what is to be measured and when. In other words, the new
Schrodinger equation must depend upon the form of all the
apparatus involved, including the measuring instruments and, for
example, whatever (or whoever) decides on the direction for a spin
determination. The work that we have outlined above suggests that
theories of this type might be possible, but much work remains to
be done and there is a danger that what emerges will look more like
an arbitrary prescription to obtain the results than like a proper
theory. Certainly it is hard to see how it can look at all natural.
There are three other points which might be relevant to this
section and which certainly should be mentioned. First, all real
measuring instruments are macroscopic. To appreciate how different
such an object is from a single electron, say, we should
realise that an object with a mass of one kilogram contains about
lo2’ particles. It is, therefore, not hard to imagine that effects which are utterly negligible for single particles might build up to
something important for macroscopic objects. Two particular ways
in which the mass of an object might appear in the formulae for
reduction are suggested at the end of Appendix 7.
Secondly, as we have seen in the previous section, all tests of
interference effects refer to particles. It is just not possible to test
whether they would also occur for macroscopic objects where a
very large number of degrees of freedom are involved. The
difference between whether they really do occur, as predicted by
quantum theory, or whether they do not, has no obvious
measurable consequences. This is unfortunate, because the question
has enormous relevance to the issues we are discussing.
Finally, if it is true that really new effects arise for large,
complex, systems, then we should ask whether there are other
manifestations of these. Is it even possible that one such effect
could be consciousness, which might also be expected to occur only
for large systems? Maybe, somewhere here, there is a link between
this section and the subject of our next chapter.