Does the human body deviate from the laws of physics, as
gleaned from the study of inanimate nature? The traditional answer
to this question is ‘No’: the body influences the mind but the mind
does not influence the body. Yet at least two reasons can be given
to support the opposite thesis.?
Here we shall examine more closely the possibility that external
reality consists of a wavefunction and that this wavefunction only
reduces when an observation is made by a conscious observer. It is
the existence of consciousness that introduces the probabilistic
aspects into the quantum world.
One reason for at least considering such a suggestion is that
wavefunction reduction (which is an inexplicable phenomenon
associated with quantum theory) and decision making (which is an
inexplicable phenomenon associated with the conscious mind)
share the common feature of producing an increase of information;
in both something previously ‘unknown’ becomes ‘known’. More
specific motivation arise, as we have seen, from the following facts.
‘Simple’ systems apparently do not reduce wavefunctions; they are
also not conscious (we ignore here the possibility (c) of the previous
section). At some stage of ‘complexity’, and for reasons that may
be solely due to complexity or may involve something totally new,
there are systems that do reduce wavefunctions, and there are
systems that are conscious. In both cases our certainty here is due
to our own experience. We experience effects which appear to
correspond to reduced wavefunctions and we are certainly
conscious. It is therefore reasonable that we should try to relate the
two phenomena, an idea that has been argued most convincingly,
in recent times, by the eminent theoretical physicist, Eugene
Wigner. (It should be noted, however, that Wigner’s latest work on
this topic shows a move to the more conventional position that
other complex, yet not conscious, systems can also cause wavefunction
reduction. We refer, for example, to his article in
Quantum Optics, Experimental Gravity, and Measurement Theory
ed P Meystre and MO Scully [New York: Plenum 19831 .)
We must now explain carefully what is involved here. We
consider an isolated system, which may be as complicated as we
desire, but which must not contain any conscious mind. According
to our assumption, such a system is described by a wavefunction
which changes with time according to the rules of quantum theory.
A conscious observer now makes a measurement of some property
of this system, e.g. of the position of a particle. When the result
of this measurement enters the mind of the observer, then the
wavefunction reduces to the form corresponding to the particular
value of the measured quantity. Notice that it is not enough for the
conscious observer simply to be aware of only the part of the
wavefunction corresponding to the observed value. If this was all
that happened then we could not be sure that a different observer
would see the same value of the observed quantity. We require that
the act of conscious observation actually changes the wavefunction.
Thus, in our potential barrier experiment (see figures 15
and 16), if an observer sees the right-hand detector as being ON
then it must be ON, and not in the state of part ON and part OFF.
Then another observer will also see that it is ON, and both will
agree that the particle has been reflected. In figure 18 we illustrate
this difference between the observed and unobserved systems.
The last paragraph shows the first of the reasons noted by
Wigner, in the quotation at the start of this section, for believing
that mind affects physical things. The second reason he gives is that
in all other parts of physics, action and reaction occur together, i.e.
if A affects B then B affects A. Thus, since the physical world
clearly affects the conscious mind, we expect the converse to apply. To be quite fair, the assertion we have made here, that conscious
minds change the wavefunction, is not absolutely necessary. It
would, presumably, be possible that the act of observation does not
change the system, but that conscious observers somehow communicate
with each other so that they all ‘see’ the same thing. Such
an interpretation of quantum theory is possible and we shall discuss
it further in 54.5. We return to our assumption that only conscious observers can
reduce wavefunctions, and must now comment on how utterly
outrageous such a statement really is. To see this we might suppose
that the detectors in the potential barrier experiment are
photographic plates. What we are saying is that they are in a state
of ‘perhaps blackened but perhaps not’ until they have been
observed by a conscious mind, which may, of course, be years after
the event. Indeed, nothing ever really happens, e.g. no particle ever
decays, except through the intervention of a conscious mind. We
are not quite in the situation of denying external reality-which
possibility we considered and rejected in 0 1.2-but we are denying
that the external world possesses the properties we observe, until we
actually observe them. This is a picture of reality that we find hard
to accept.
The paradox of ‘Schrodinger’s cat’ is an example of the sort of
problem we can get into here. We suppose, for example, that the
right-hand detector in our potential barrier experiment is a trigger
that fires a gun and kills a cat as soon as a particle reaches it. After
one particle has passed through the apparatus the wavefunction
thus contains a piece in which the cat is dead and a piece in which
the cat is alive. Only if the cat is conscious can we say that one of
these represents the truth. What however could we say if the cat
were asleep? If, on the other hand, a cat is not conscious, or if we
used instead a being or a thing that is not conscious, then it remains
in a state of being part-dead/part-alive until some conscious
observer forces the wavefunction to go to one state or the other.
Like Schrodinger himself we probably consider this an unlikely
picture of reality.
The assumption we are considering appears even more weird
when we realise that throughout much of the universe, and indeed
throughout all of it at early times, there were presumably no conscious
observers. Thus the wavefunction did not reduce, and all the
possibilities inherent in the development of the wavefunction since
the beginning of time would have persisted until the first conscious
observers appeared. Even worse are the problems we meet if we
accept the modern ideas on the early universe in which quantum
decays (of the ‘vacuum’, but this need not trouble us here) were
necessary in order to obtain the conditions in which conscious
observers could exist. Who, or what, did the observations necessary
to create the observers? The only possibility here seems to be that observation, indeed
conscious observation, can be made by ‘minds’ outside the physical
universe. Such is one of the traditional roles of God and/or gods.
This is the realm of theology; a realm into which we shall, with
trepidation, enter briefly in the next section.
Before we close this section, however, there is one obvious
question we must ask. Since we have suggested that consciousness
might offer a possible, even if unlikely, solution to a problem of
physics, can physics help with the problem of the nature of consciousness?
Again the answer may well be that it cannot, but the
issue is certainly being discussed. The fact that quantum theory
frees physics from the rigid causality of classical mechanics is an
obviously immediately relevant fact. There just seems to be more
room for ideas like free will in a quantum world than in a classical
one. Already quantum tunnelling, as described in 81.3, has been
used to explain certain processes in the nervous system-see, for
example, Walker, International Journal of Quantum Chemistry 11
103 (1977). (We should, however, be cautious here. There is a big
difference between the idea of freedom to choose, where the choice
is presumably made by rational thought, and the apparent
randomness of quantum theory, so a relation between the two,
though possible, is not obvious,) It is also natural to try to associate
the very non-local nature of wavefunctions with the similar lack of
locality of ‘thoughts’, etc. For some discussion along these lines,
and for other references, we refer to the article by Stapp,
‘Consciousness and Values in the Quantum Universe’, Foundations
of Physics 15 35 (1985).
Monday, July 2, 2007
What is a conscious observer?
Consciousness or ‘awareness’ is something we, as people, possess.
We talk about it; we have a vague understanding of what it is;
through it we experience many emotions, happiness, sorrow,
jealousy, love, etc; we develop concepts like free will and purpose
which really have no meaning without it; we can even refer to its
absence, e.g. to ‘unconscious’ decisions, etc; but we do not have any way of defining it. It cannot be expressed in terms of other
things or even be likened to other properties. It is something unique
and totally different from anything else.
To discuss it, we should therefore begin with what we know. Or,
rather, I should begin with what Z know.
I am conscious. This fact I can express in an alternative way by
saying that I have a conscious mind, or that my consciousness
exists, However expressed, and regardless of the fact that we do not
really know precisely what the statements mean, the truth they
convey cannot be denied, Even if I wish to deny reality to
everything else, which, as we saw in 0 1.2, is logically possible even
if rather pointless, I cannot deny the reality of my own thoughts.
As a natural extrapolation of my experience, it is reasonable that
I should assume that you, my readers, are conscious, and then to
extend this to all other people. Already, however, there are those
who would question this. A Princeton University psychologist,
Janes, has written a book in which he claims that consciousness is
a comparatively recent feature of the human race. (The book is
called The Origin of Consciousness in the Breakdown of the
Bicameral Mind [ Harmondsworth: Penguin 19801 . Though I am
fairly convinced that I do not believe the claim, it is expertly, and
interestingly, argued.)
Having agreed that we possess consciousness, do we know what
it is? It is a private ‘space’ in which each of us rules alone, and into
which we can introduce whatever we desire of real things, i.e. those
we believe exist elsewhere, or abstract things which are purely our
creation. But does such a vague description allow us to say where
else it might exist, who, or what, might possess it? People? Yes, by
extension of ourselves. But dogs? worms? amoebae?
I hope my readers will allow me a personal note here. I
remember, as a schoolboy, sitting by a riverside listening to a
skylark. I think I should have been revising for examinations but,
instead, I doodled some verses of poetry. Though I can just about
remember them, they are inferior to the precedent I was following,
so I will not expose them to public view. I mention them because
in the first verse I asked whether the bird was singing because it was
happy, in the second I wondered whether it was instead singing in
response to feelings of sadness, but then I wondered whether it was
neither, whether in fact the skylark was capable of feeling either
happy or sad or whether it possessed any awareness of anything:
‘Is’t only nature’s law that makes thee want to sing?’ I was asking
myself, perhaps for the first time, the question about consciousness
that I have asked many times since. I still do not know the answer,
and I have no idea how to go about finding it.
The nature of the problem here can be demonstrated by the
following thought experiment (which could, with a little expense,
even be a real experiment). Suppose we devised a series of tests for
consciousness. A conscious being would, for example, be expected
to show pleasure in some suitable way when it was praised, it would
back away from any object that hit it, or otherwise showed
threatening behaviour, it would seek ‘food’, i.e. whatever it
required to sustain its activity, when needed, and would express the
need urgently if the search proved unsuccessful. The list could easily
be extended. Whatever property of this type we included, however,
it is easy to see that we could design a computer-robot to make all
the correct responses. Such a machine would pass our tests for
consciousness. I believe, though I am not sure why, that it would
nevertheless not be conscious. Somehow ‘physical’ systems, even
when designed to have the attributes of consciousness, do not seem
to us to be conscious. Thus, although it is easy to simulate the
effects of consciousness, we should avoid making the mistake of
believing that in so doing we have created consciousness.
Conversely, it would be possible, by careful analysis of what
happens in the human brain, to correlate the various feelings like
joy, sadness, anger, etc, which we associate with consciousness,
with particular chemical or physical processes in the body, the
release of various hormones, and such like. But surely joy is not a
chemical compound, or a particular pattern of electrical currents,
Or is it? Or is it just caused by particular physical processes
occurring in the right place?Alternatively, is the truth nearer to the
statement that the thoughts of the conscious mind cause the
appropriate currents to flow? Are the emotions, or their material
effects, primary?
Certainly conscious thoughts appear to have physical effects. I
have just made a conscious decision to write these particular words
in my word processor. The fact that you are reading them is
evidence that my thoughts had real effect in the physical world. In
one sense, of course, this could be an illusion (whatever that might
mean in this context). The process of my writing these words could
be entirely a consequence of all the particles that make up my hand, brain, etc, moving inexorably according to the laws of motion.
Somewhere along the series of events in my body that leads to the
typing, particular things happen that make me think I am
‘deciding’ what to write. But what is cause, and what merely effect?
The problems we are discussing here are more than simply a
question of the language that we use to describe things. There are
of course ‘language’ issues. For example, we could describe a
pocket calculator as a machine that ‘allows particular currents to
flow’, or, alternatively, as a machine that ‘does arithmetic’. These
are different sets of words describing the same thing. Our concern
is more with the question of whether the calcualator knows that it
is doing anything at all. That real issues are involved can be seen
from the fact that our behaviour to some extent depends on how
we answer these questions. Part of the reason for the concern we
sometimes (too rarely) feel for people, animals, . . . is that we
believe these creatures are conscious.
It is outside the range of this particular book, and beyond the
ability of its author, to take this discussion any further. Much has
been written on the subject. In this sense it is rather like the interpretation
problem of quantum theory. I have the impression that
the two topics are similar in another sense-very little is understood
of either!
We close this section by offering three possible ‘answers’ to the
question of what makes an object conscious.
We talk about it; we have a vague understanding of what it is;
through it we experience many emotions, happiness, sorrow,
jealousy, love, etc; we develop concepts like free will and purpose
which really have no meaning without it; we can even refer to its
absence, e.g. to ‘unconscious’ decisions, etc; but we do not have any way of defining it. It cannot be expressed in terms of other
things or even be likened to other properties. It is something unique
and totally different from anything else.
To discuss it, we should therefore begin with what we know. Or,
rather, I should begin with what Z know.
I am conscious. This fact I can express in an alternative way by
saying that I have a conscious mind, or that my consciousness
exists, However expressed, and regardless of the fact that we do not
really know precisely what the statements mean, the truth they
convey cannot be denied, Even if I wish to deny reality to
everything else, which, as we saw in 0 1.2, is logically possible even
if rather pointless, I cannot deny the reality of my own thoughts.
As a natural extrapolation of my experience, it is reasonable that
I should assume that you, my readers, are conscious, and then to
extend this to all other people. Already, however, there are those
who would question this. A Princeton University psychologist,
Janes, has written a book in which he claims that consciousness is
a comparatively recent feature of the human race. (The book is
called The Origin of Consciousness in the Breakdown of the
Bicameral Mind [ Harmondsworth: Penguin 19801 . Though I am
fairly convinced that I do not believe the claim, it is expertly, and
interestingly, argued.)
Having agreed that we possess consciousness, do we know what
it is? It is a private ‘space’ in which each of us rules alone, and into
which we can introduce whatever we desire of real things, i.e. those
we believe exist elsewhere, or abstract things which are purely our
creation. But does such a vague description allow us to say where
else it might exist, who, or what, might possess it? People? Yes, by
extension of ourselves. But dogs? worms? amoebae?
I hope my readers will allow me a personal note here. I
remember, as a schoolboy, sitting by a riverside listening to a
skylark. I think I should have been revising for examinations but,
instead, I doodled some verses of poetry. Though I can just about
remember them, they are inferior to the precedent I was following,
so I will not expose them to public view. I mention them because
in the first verse I asked whether the bird was singing because it was
happy, in the second I wondered whether it was instead singing in
response to feelings of sadness, but then I wondered whether it was
neither, whether in fact the skylark was capable of feeling either
happy or sad or whether it possessed any awareness of anything:
‘Is’t only nature’s law that makes thee want to sing?’ I was asking
myself, perhaps for the first time, the question about consciousness
that I have asked many times since. I still do not know the answer,
and I have no idea how to go about finding it.
The nature of the problem here can be demonstrated by the
following thought experiment (which could, with a little expense,
even be a real experiment). Suppose we devised a series of tests for
consciousness. A conscious being would, for example, be expected
to show pleasure in some suitable way when it was praised, it would
back away from any object that hit it, or otherwise showed
threatening behaviour, it would seek ‘food’, i.e. whatever it
required to sustain its activity, when needed, and would express the
need urgently if the search proved unsuccessful. The list could easily
be extended. Whatever property of this type we included, however,
it is easy to see that we could design a computer-robot to make all
the correct responses. Such a machine would pass our tests for
consciousness. I believe, though I am not sure why, that it would
nevertheless not be conscious. Somehow ‘physical’ systems, even
when designed to have the attributes of consciousness, do not seem
to us to be conscious. Thus, although it is easy to simulate the
effects of consciousness, we should avoid making the mistake of
believing that in so doing we have created consciousness.
Conversely, it would be possible, by careful analysis of what
happens in the human brain, to correlate the various feelings like
joy, sadness, anger, etc, which we associate with consciousness,
with particular chemical or physical processes in the body, the
release of various hormones, and such like. But surely joy is not a
chemical compound, or a particular pattern of electrical currents,
Or is it? Or is it just caused by particular physical processes
occurring in the right place?Alternatively, is the truth nearer to the
statement that the thoughts of the conscious mind cause the
appropriate currents to flow? Are the emotions, or their material
effects, primary?
Certainly conscious thoughts appear to have physical effects. I
have just made a conscious decision to write these particular words
in my word processor. The fact that you are reading them is
evidence that my thoughts had real effect in the physical world. In
one sense, of course, this could be an illusion (whatever that might
mean in this context). The process of my writing these words could
be entirely a consequence of all the particles that make up my hand, brain, etc, moving inexorably according to the laws of motion.
Somewhere along the series of events in my body that leads to the
typing, particular things happen that make me think I am
‘deciding’ what to write. But what is cause, and what merely effect?
The problems we are discussing here are more than simply a
question of the language that we use to describe things. There are
of course ‘language’ issues. For example, we could describe a
pocket calculator as a machine that ‘allows particular currents to
flow’, or, alternatively, as a machine that ‘does arithmetic’. These
are different sets of words describing the same thing. Our concern
is more with the question of whether the calcualator knows that it
is doing anything at all. That real issues are involved can be seen
from the fact that our behaviour to some extent depends on how
we answer these questions. Part of the reason for the concern we
sometimes (too rarely) feel for people, animals, . . . is that we
believe these creatures are conscious.
It is outside the range of this particular book, and beyond the
ability of its author, to take this discussion any further. Much has
been written on the subject. In this sense it is rather like the interpretation
problem of quantum theory. I have the impression that
the two topics are similar in another sense-very little is understood
of either!
We close this section by offering three possible ‘answers’ to the
question of what makes an object conscious.
The relevance of conscious observers
In $3.5 we proved that an instrument governed by the laws of
quantum theory is not capable of making a proper measurement,
that is, it cannot cause the wavefunction of a system to change to
a state corresponding to a particular value of the quantity to be
measured. As an example we saw that, in the potential barrier
experiment, even after the attempted measurement of transmission
or reflection, the wavefunction still contained pieces corresponding
to both possibilities.
We must not, of course, conclude from this that true
measurements are impossible. We know that they occur. We can
observe which of the two detectors flashes and hence deduce
whether or not a particle has passed through the barrier. Our brain
certainly does not permit both possibilities. Thus, although a
simple, microscopic, instrument, obeying the laws of quantum
theory, does not reduce wavefunctions, they are certainly reduced
by the time the information reaches our brain.
What, then, is responsible for the reduction and what are the
characteristics of ‘instruments’ that are able to cause it? We do not
know the answers to these questions. It could be that, with increasing
complexity and size, correction terms in the equations of quantum
mechanics become more significant, so that any macroscopic
apparatus can do the reduction. On the other hand, it could be that
something totally new is required and that some things possess it whereas others do not. In either case it is an obvious question to
ask whether there are other features of wavefunction-reducing
systems that distinguish them from simpler systems. One obvious
possibility that arises here is to go to the extreme end of the chain
of observation and consider the possibility that the reduction does
not occur until we know that it must, i.e. that it only occurs when
conscious observers are involved.
Such a wild suggestion tends to horrify the austere minds of
most physicists. We fear that it takes our subject, beloved for its
high standards of objectivity, rigour, precision and experimental
support, into a realm where nothing can be properly defined, where
feelings and personality replace detached measurement, even, perhaps,
to put it on a par with astrology and the reading of tea leaves!
From another point of view, however, it should perhaps be seen as
an exciting new development. Maybe it allows the possibility that
the enormously successful methodology of physics might enter a
totally new field of investigation. This would be a revolution that
would, in its significance, dwarf those to which we referred in 4 1.1.
Although it is probably fair to say that such a revolution is
unlikely, we should, before dismissing it entirely, remember that
J C Maxwell, the creator of the theory of electromagnetism and
undoubtedly one of the greatest physicists of all time, once
expressed the view that the study of atoms would be forever outside
the scope of physics! Such a precedent will guard us from making
similar rash statements about consciousness.
If we are to consider seriously the relevance of consciousness in
the collapse of wavefunctions we must ask, and at least try to
answer, the question of what it is. To this topic we turn in our next
section.
quantum theory is not capable of making a proper measurement,
that is, it cannot cause the wavefunction of a system to change to
a state corresponding to a particular value of the quantity to be
measured. As an example we saw that, in the potential barrier
experiment, even after the attempted measurement of transmission
or reflection, the wavefunction still contained pieces corresponding
to both possibilities.
We must not, of course, conclude from this that true
measurements are impossible. We know that they occur. We can
observe which of the two detectors flashes and hence deduce
whether or not a particle has passed through the barrier. Our brain
certainly does not permit both possibilities. Thus, although a
simple, microscopic, instrument, obeying the laws of quantum
theory, does not reduce wavefunctions, they are certainly reduced
by the time the information reaches our brain.
What, then, is responsible for the reduction and what are the
characteristics of ‘instruments’ that are able to cause it? We do not
know the answers to these questions. It could be that, with increasing
complexity and size, correction terms in the equations of quantum
mechanics become more significant, so that any macroscopic
apparatus can do the reduction. On the other hand, it could be that
something totally new is required and that some things possess it whereas others do not. In either case it is an obvious question to
ask whether there are other features of wavefunction-reducing
systems that distinguish them from simpler systems. One obvious
possibility that arises here is to go to the extreme end of the chain
of observation and consider the possibility that the reduction does
not occur until we know that it must, i.e. that it only occurs when
conscious observers are involved.
Such a wild suggestion tends to horrify the austere minds of
most physicists. We fear that it takes our subject, beloved for its
high standards of objectivity, rigour, precision and experimental
support, into a realm where nothing can be properly defined, where
feelings and personality replace detached measurement, even, perhaps,
to put it on a par with astrology and the reading of tea leaves!
From another point of view, however, it should perhaps be seen as
an exciting new development. Maybe it allows the possibility that
the enormously successful methodology of physics might enter a
totally new field of investigation. This would be a revolution that
would, in its significance, dwarf those to which we referred in 4 1.1.
Although it is probably fair to say that such a revolution is
unlikely, we should, before dismissing it entirely, remember that
J C Maxwell, the creator of the theory of electromagnetism and
undoubtedly one of the greatest physicists of all time, once
expressed the view that the study of atoms would be forever outside
the scope of physics! Such a precedent will guard us from making
similar rash statements about consciousness.
If we are to consider seriously the relevance of consciousness in
the collapse of wavefunctions we must ask, and at least try to
answer, the question of what it is. To this topic we turn in our next
section.
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.
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.
Interference and macroscopic objects
We have stressed that a wavefunction which contains a sum of
several terms (a pure state) is genuinely different from a wavefunction
which is either one term or another (a mixed state).
The reason for the difference is that, in principle, it is possible
to arrange that two terms in a sum interfere when a particular probability is calculated, and this interference can be observed.
Indeed, as we have seen, e.g. in $2.5, there are many experiments
where this interference has been measured and found to agree
perfectly with the predictions of quantum theory. However, in
practice, in many situations, and in all cases where macroscopic
apparatus is involved, it is not possible to design a suitable experiment
to observe the interference, so the two wavefunctions are
effectively indistinguishable.
To understand why this is so, let us suppose we want to check
that the wavefunction for the barrier-plus-two-detectors experiment
really does contain the sum of two pieces, e.g. as in figure
15(b). To this end we would like to arrange that the two pieces are
allowed to interfere. Thus, in effect, we need to do both the potential
barrier experiments of Chapter One in the same experiment.
However, even when the waves corresponding to the reflected and
transmitted particles are brought together by mirrors they will not
interfere because, unlike the situation in 01.4, they now contain
different states of the detectors (ON/OFF or OFF/ON respectively). In
order to have interference it is necessary that the detectors be
brought to the same state. At first sight this might appear to be
easy; they can be switched to the OFF position, say. However, in
order to have interference the states must be identical, and for
macroscopic objects that is not possible. To reverse exactly the
process whereby one of the detectors was switched to ON is, by
many orders of magnitude, outside the range of any conceivable
experimental technique; there is, for example, no conceivable
mechanical interaction between macroscopic objects that does not
remove a few atoms, slightly change the temperature of the object,
alter its shape, etc. This is the reason why interference between
macroscopic objects cannot be experimentally verified.
For a proper treatment of this topic we would need to use the
mathematical formalism of quantum mechanics. Some of the ideas
are discussed further in Appendix 6. Here we shall be content with
the above rather sketchy outline of the argument. The key to it, to
which we shall return, is the inherently irreversible nature of
macroscopic changes.
It is clear that there is a continuum of scales ranging from the
micro- to the macroscopic, so we naturally ask how far towards the
latter we can go with interference experiments. At present, it seems
as though the answer is not very far: all experiments so far performed deal with ‘elementary’ particles, or, more precisely, with
systems that, for the purpose of the experiment considered, can be
regarded as having very few degrees of freedom. Some ideas for
doing interference experiments with larger systems are being
explored at the present time
several terms (a pure state) is genuinely different from a wavefunction
which is either one term or another (a mixed state).
The reason for the difference is that, in principle, it is possible
to arrange that two terms in a sum interfere when a particular probability is calculated, and this interference can be observed.
Indeed, as we have seen, e.g. in $2.5, there are many experiments
where this interference has been measured and found to agree
perfectly with the predictions of quantum theory. However, in
practice, in many situations, and in all cases where macroscopic
apparatus is involved, it is not possible to design a suitable experiment
to observe the interference, so the two wavefunctions are
effectively indistinguishable.
To understand why this is so, let us suppose we want to check
that the wavefunction for the barrier-plus-two-detectors experiment
really does contain the sum of two pieces, e.g. as in figure
15(b). To this end we would like to arrange that the two pieces are
allowed to interfere. Thus, in effect, we need to do both the potential
barrier experiments of Chapter One in the same experiment.
However, even when the waves corresponding to the reflected and
transmitted particles are brought together by mirrors they will not
interfere because, unlike the situation in 01.4, they now contain
different states of the detectors (ON/OFF or OFF/ON respectively). In
order to have interference it is necessary that the detectors be
brought to the same state. At first sight this might appear to be
easy; they can be switched to the OFF position, say. However, in
order to have interference the states must be identical, and for
macroscopic objects that is not possible. To reverse exactly the
process whereby one of the detectors was switched to ON is, by
many orders of magnitude, outside the range of any conceivable
experimental technique; there is, for example, no conceivable
mechanical interaction between macroscopic objects that does not
remove a few atoms, slightly change the temperature of the object,
alter its shape, etc. This is the reason why interference between
macroscopic objects cannot be experimentally verified.
For a proper treatment of this topic we would need to use the
mathematical formalism of quantum mechanics. Some of the ideas
are discussed further in Appendix 6. Here we shall be content with
the above rather sketchy outline of the argument. The key to it, to
which we shall return, is the inherently irreversible nature of
macroscopic changes.
It is clear that there is a continuum of scales ranging from the
micro- to the macroscopic, so we naturally ask how far towards the
latter we can go with interference experiments. At present, it seems
as though the answer is not very far: all experiments so far performed deal with ‘elementary’ particles, or, more precisely, with
systems that, for the purpose of the experiment considered, can be
regarded as having very few degrees of freedom. Some ideas for
doing interference experiments with larger systems are being
explored at the present time
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.
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.
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.
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.
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