Hi everyone! Firstly, many apologies for this being late.
Since we never got around to lecture 5, I have focused this
post on the end of lecture 4, namely the third proposed single world
interpretation of quantum mechanics: the dynamical collapse theory.
I thought I’d summarise my understanding of this general
interpretation, before focusing on a few specific examples, highlighting any
problems that I had in interpreting them. Here goes…
Objective Collapse theories, also known as QMSL theories
(Quantum Mechanical Spontaneous Localisation), are realistic, indeterministic
and fundamentally reject hidden variables. The general idea is that the wave
function associated with all sub-atomic particles has some ontological reality,
along with a very small but finite probability of collapse. The time scale
associated with the collapse of individual particles is thought to be of the
order of 108 years. When an
observer makes a measurement using some physical piece of apparatus (be that an
eye, or lab equipment etc) then the observed particle becomes entangled with
the observing equipment, which will necessarily be a very large collection of
atoms >108. The chance of any one of the large collection of
particles collapsing is now very high, and since they are entangled, if one
collapses so too must all the particles in the system.
Dynamical collapse theories originated out of attempts to
get around the measurement problem in quantum mechanics, which inexplicably
gives the observer in any system a special position, in that they are solely
responsible for the collapse of said system. Such a view, as held in the
popular ‘Copenhagen Interpretation’, leaves much room for interpretation in
what constitutes an observer or what qualifies as an observation. DCTs neatly
remove this issue altogether by attributing the collapse of the wave function
to random processes with no contribution by the observer other than providing
the measuring equipment.
**As an aside, I was wondering how you might justify the
view that particles just so happen to follow a theory of random collapse, but
after further thought I realised that there are many examples in physics of
this, spontaneous atomic decay in nuclear fission being the most obvious
example. I suppose this view still suffers from the issue that Einstein took with
QM (god does not play dice) in that it offers no explanation for the basis of
the random collapse in the first place, to which the only retort I am aware of,
is that this is just fundamentally how the universe operates, and we as humans
are not in a position to suppose it should be any other way. Incidentally if
anyone has anything to say on this then I would be keen to discuss, as I don’t
find it altogether that satisfying.**
Many theories fall under the dynamical collapse umbrella,
but as far as I am aware there are two main variations, based on the mechanics
of the wave-function collapse:
1.
In the first group the collapse is found
‘within’ the wave-function. I.e. the formula which guides the evolution of the
system (the Schrodinger eq), which under normal formulation describes the state
of the system as some linear superposition of the basis states, is adjusted in
some non-linear manner. This adjustment means the governing equation actually
describes the collapse itself. The original GRW paper is a good example of this
view.
2.
Evolution of the wave function remains
unchanged, with the additional collapse process added in. The most famous
example would be the ‘Penrose Interpretation’.
The first variation is the most accepted version of the
theory I believe, although it has been modified a little since it’s first
inception by GRW in the 80’s. The original paper didn’t respect the symmetries
of many particle systems (didn’t conserve parity?), so a modified version of
the theory was created, known as CSL (Continuous Spontaneous Localisation),
which amends the shortcomings mentioned above.
Despite the many attempts to reconcile dynamical collapse
with our present understanding of QM, there remains a fundamental issue with
variation 1). Since the collapse occurs from within the wave function, the
collapse cannot be complete or else the principle of conservation of energy
would be broken. Why this must be is not completely clear to me, so I would
like to discuss this tomorrow if possible. A non-complete collapse implies that
the wave function has some non-zero amplitudes for states not described by the
collapse, and so in theory there is a non zero chance of the system (or
particle) simultaneously jumping to another state. This is fairly
counter-intuitive and has not been verified by experiment thus far.
I will now discuss the interpretation of the second
variation of DCT, namely the ‘Penrose Interpretation’. Penrose theorises that
the wave-function collapses to it’s basis states when the space-time curvature
of the quantum states attains a significant level, which he speculates to be
around 1 graviton, (the hypothesised but yet to be discovered force carrying
particle in quantum gravity).
To explain this, let’s consider a solitary particle, which
has multiple basis states that it could occupy (eg the position of an electron
around an atom). The theory states that each of the electron’s states exists
simultaneously, and so the electron creates a gravitational field in all the
positions that it can exist according to its basis states. A field
intrinsically contains energy, and requires energy to be maintained, thus the
larger the mass (energy) of the particle, the higher the required energy that
is needed to maintain all the fields that exist curtsey of all the
simultaneously existing states. The next step is to say that for a small
particle, the energy required is small enough that it can exist as a
superposition indefinitely. (I couldn’t find a precise reason for this, but I
think you can use the uncertainty principle ∆E∆t>h/2π and say that for a
very small mass, the time can be extremely large).
Likewise for larger particles, their super positions require
too much energy to simultaneously exist for any significant period of time
(again, presumably found from the U-P). Since it would require less energy to
maintain one gravitational field than all the countless super positions, Penrose
believes that this energy requirement causes larger mass systems to collapse to
a single state, with the probability of collapsing to a specific basis state
given by the square of the amplitude as usual. Penrose speculates that the
transition from macroscopic to the sub-atomic domain would happen at around the
mass of a dust particle.
To summarise, both variations of DCT attractively side step
the measurement problem, giving no special position to an observer. They also rule
out many world theories by holding the collapse to curtail the branching of the
wave function, removing unobserved behaviour.
Discussion Points:
Personally I find the Penrose interpretation to be the most
attractive of all the interpretations we have seen so far. The Copenhagen interpretation
throws up countless issues with the interpretation of what the wave function
represents and why observers should be gifted such a special position in the
universe.
Variation 1. of the DCT seems a little like GRW have moulded
the solution to fit the answer. I don’t like resorting to spontaneous and
randomly collapsing wave functions, but as I mentioned previously there are
other example in physics of this kind of thing.
I am also rather unclear on how the wave function can have
any physical reality, and what that would mean or look like.
If you’ve got to the bottom of this then thanks for staying
with me, I’m sure some of this is quite unclear so looking forward to discussing
tomorrow!
Best,
Jamie
RE: energy conservation in GRW-style collapses
ReplyDeleteI'm not entirely sure what you mean by `complete' collapses, it seems like you mean that during the collapse process the wavefunction collapses to a delta-function ie a `state with definite position' or or during collapse the wavefunction is multiplied by an `infinitely thin Gaussian' (then normalised). But collapses such as these aren't allowed regardless of any considerations of conservation of energy---they aren't allowed because a delta-function wavefunction isn't a permissible state (it isn't square-integrable).
On the other hand, I think that energy isn't conserved (in general) during GRW-style collapses (ie multiplication by a [position basis] Gaussian then normalisation) regardless of the width of the Gaussian.
Consider the following example: a particle is initially in the groundstate of some potential well. It undergoes a GRW-style collapse; now, in general, its wavefunction will have changed. The wavefunction must be expressible as a sum of energy eigenstates. Since the wavefunction has changed the possible outcomes of an energy measurement must have changed (from only the groundstate energy to the energies of whatever eigenstates have non-zero amplitudes in the resulting expansion) and furthermore since it was in the groundstate to start with the average energy must have increased.
Hi Jon, thanks for raising some interesting points.
ReplyDeleteBy 'complete' collapse I was referring to a collapse that does not have the probability of 1 associated with any one basis state. That is to say, there is a non-zero chance that the particle will jump to a state other than that to which it originally collapsed. Whether the jump can occur simultaneously under the theory, or whether it requires a re-measurement I am unclear about.
In your first paragraph you seem to be suggesting that there is no such thing as complete collapse in any theory. I'm afraid I don't understand your claim that a delta function is not allowed because it 'isn't square integrable'. A delta function, by definition, will yield 1 if you integrate over the function. The very reason delta functions are used here is that they give a precise position whilst maintaining an integrable area of 1, therefore giving you the probability of 1 to find the particle in the specific state corresponding to the peak of the delta. In my understanding, let's say we measure the position of a particle, thereby collapsing the wave function in the position basis, the only function that gives a definite probability of finding the particle in the state that it has collapsed to is indeed the delta function. So I’m not really sure what you meant by them not being allowed, but maybe I misunderstood you.
Your point that energy is never conserved in GRW collapses is an interesting one, although I have a few problems accepting your example. You said that the particle is initially in the ground state, but to make such a claim, surely we would have to measure the particle in the first place. Before measurement we surely can't say what state the particle is in, hence the existence of the wave function and the superposition of the ground state and all the energy states forming said wave function. Your example goes onto suggest that the wave function actually moves from being a specific state (the ground state) to becoming some super position of all the energy states, through the process of collapse, but this is the wrong way around. The eigenfunction of the particle goes from being a superposition of the energy states, to being a specific state upon collapse. I think your main point still stands though: if the wave function goes from a general superposition to being a specific state, in all likelihood it has changed in energy. Actually saying that, position and energy are commuting operators, therefore I’m not convinced the wave function would change at all, since the measurement of position should have no bearing on the measurement of energy.
If the conclusion of this is that no collapse can occur without violating the conservation of energy, then I wonder how we can entertain the theory of collapse in the first place...?
Thinking further about the issue that energy doesn't appear to be conserved if you collapse the wave function, I think there is a way out. I think a good argument that energy is indeed conserved would say that by measuring the system, we have interacted with it in some meaningful way. This interaction is precisely what has changed the energy of the system. An example would be measuring the energy of a particle by firing photons at it. The photons have necessarily hit the particle we are looking at, and thus influenced it's energy. Thus the energy of the particle may well be conserved, but we will never know since we are only viewing the particle after it's bombardment by our photons.
ReplyDeleteThanks for your post Jamie. From the literature I also found the Penrose interpretation to be the most compelling, and I agree that it seems to provide the most comprehensive account without side stepping some pretty significant difficulties! I was slightly confused by your comment on the conservation of energy though, and to be honest I can't quite follow your discussion with Jon about it. Maybe this will become clearer in class :)
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