Measurement in quantum theory

As we have seen, it is not normally correct to say that a particle,
described by quantum theory, is at a particular position. Rather,
the particle has a wavefunction which tells us the probability of
finding it at any given position when a measurement of position is
made. Similarly, the wavefunction tells us the probability of
obtaining a given value for the velocity if we make a velocity
measurement. Thus measurements play a more positive role in
quantum theory than in classical physics because they are not
merely observations of something already present, they actually
help to produce it.
A measuring instrument can be defined as something that
enables us to make a measurement of the above type. That such
instruments exist follows from the fact that we do actually make
such measurements. We would, of course, like to believe that the
apparatuses can be described by physics, i.e. that they too satisfy
the rules of quantum theory. It is, however, very easy to show that
this is impossible. An instrument that is able to make a measurement,
in the above sense, cannot be completely described by
quantum theory.
To illustrate this fact we shall consider again the potential barrier
experiment with the two detectors in position. Recall that the lefthand
detector records the passage of a transmitted particle and the
right-hand detector the passage of a reflected particle. We suppose
that each detector is a simple quantum mechanical system that can exist in one of two states OFF and ON, and that the transition
between these is caused by the passage of a particle through the
detector.
The complete experiment is now described by a wavefunction
which contains information about both detectors as well as about
the particle. Thus, for example, it would tell us the probability of finding the particle at a given position, with one detector in the OFF
position and the other in the ON position, etc. We know the initial
form of this wavefunction; it describes the particle as being incident
from the right and both detectors being in the OFF position. A
pictorial representation of this is given in figure 15(a).
The system now evolves with time according to the Schrodinger
equation. This equation is more complicated than before because
it must include the interaction between the detectors and the
particle. We are assuming that this interaction only occurs when the
particle is in the neighbourhood of a detector, and that its effect is
to change the detector from OFF to ON as the particle passes
through. The precise details here are not important. We can then
go to a later time when the particle will certainly have passed
through one detector, i.e. the two parts of the wavefunction shown
in figure 11 have passed beyond the positions of the detectors. The
wavefunction will now be the sum of two pieces (compare the
discussion given earlier). The first piece describes a peak travelling
to the right, with the right-hand detector ON and the left-hand
detector OFF. The second describes a peak travelling to the left,
with the right-hand detector OFF and the left-hand detector ON.
Figure 15(b) gives a picture of this wavefunction.
We notice, first, that our measuring instruments are doing their
job properly in the classical sense, that is they correctly correlate
the ON/OFF positions of the detectors with the reflection/transmission
of the particle. However they have not selected one or the
other; the wavefunction still contains both possibilities and has not
been reduced. Thus we have not succeeded in making a proper
measurement in the quantum theoretical sense as we described it at
the beginning of this section. Such a measurement would have left
us with a final state expressible as either the left-hand detector ON
and the right-hand detector OFF, or the other way round (with a
certain probability) and not as the sum of both. Pictorially, the
wavefunction would have had the form of figure 16 rather than
figure 15(b). Readers who wish to see the difference expressed in
terms of mathematical symbols should consult §4.5.
It is important now to realise that the difference between these
two forms of wavefunction is not just ‘words’ (or even, in 84.5,
‘symbols’). They are different. The difference can be seen from the
fact that, at least in principle (see next section), the two parts of the
sum can be brought together and made to interfere. Such interference is not possible if the wavefunction has become just one of
the two pieces.

The result we have obtained, that quantum theory does not allow
the reduction of the wavefunction, is extremely important. We
have obtained it in a very specialised and idealised situation, but in
fact it is a completely general result. A wavefunction that can be
expressed as a sum of several terms, like that of figure 15(b), is called
a pure state. One that is expressed as a selection of alternative
possibilities, like that of figure 16, is called a mixed state. From the
laws of quantum theory it is possible to prove that a pure state cannot
change into a mixed state. Thus the wavefunction can never be
reduced. An easy way to understand why this is so is to recall that wavefunctions change with time in a deterministic way, as long as
they are described by quantum theory, hence they can never give
the probabilistic aspects associated with measurements.
Note that we cannot solve our problem by saying, in the potential
barrier example, that all we need to do is look at the detectors to
see whether they are ON or OFF. This is equivalent to saying that we
measure the state of the detectors. We then have to repeat the
process and describe the new measuring apparatus, e.g. our eyes,
by quantum theory. The resulting wavefunction now contains
information describing this additional apparatus. It will remain a
pure state.
Quantum theory, therefore, when applied to individual systems,
contains an internal contradiction. It cannot describe instruments
suitable for making measurements.
Faced with this situation, and bearing in mind the enormous success
of quantum theory, it is natural that we should seek to modify
it in such a way as to leave its successful predictions unchanged and
yet to allow wavefunction reduction in appropriate circumstances.
Attempts along these lines will be described in 43.7, and we shall
see that there are formidable problems.
Are there any alternatives? Well, if quantum theory says that
wavefunctions do not reduce we should look again at why we need
them to reduce in the first place. Why must measurements choose?
How do we know that a detector will tell us that a particle either
passed through or not? The obvious answer is that we are conscious
of seeing only one result. Our conscious minds do not contain both
parts of the wavefunction. Maybe, then, in order to understand
what is happening, we need to examine this answer more closely
and to consider the concept of consciousness.