Re: Determinacy in classical physics (naive question)

mmeron_at_cars3.uchicago.edu
Date: 07/19/04 

Date: Mon, 19 Jul 2004 03:18:01 GMT

In article <cdf3bn$hin@odbk17.prod.google.com>, "Andrew B. Park" <novakyu@yahoo.com> writes:
>mmeron@cars3.uchicago.edu wrote:
>(order changed, for continuity)
>>
>> Determinism (in physics), means that given the knowledge of a
>> sufficient set of physical parameters of a system at time t_0, the
>> values of all physical parameters of said system at any subsequent
>> time t > t_) are uniquely determined. In more physical terms this
>> means that given two systems, identically prepared, at time t_0, the
>> result of any physical measurement performed on both systems at any
>> subsequent time t are the same.
>>
>(/order)
>
>Yeah, that's what I thought. I just thought I would illustrate it with
>measurable quantities. :)
>
>(snip)
>> To begin with, there is nothing magical about either positions and
>> momenta or the uncertainty principle (and, no, the uncertainty
>> principle is *not* an axiom, it is a derived result withing quantum
>> mechanics). You've an uncertainty principle in *classical* wave
>> theory as well, nevertheless classical wave theory is fully
>> deterministic.
>> As you may notice, there is not a word in the above about position
>and
>> momenta. These happen to be a convenient "sufficient set of physical
>
>> parameters" for a classical system of particles, but other than this
>> there is nothing special about them. For other systems, other sets
>of
>> parameters may be used.
>(snip)
>
>There are a few things here that I had always been wondering about...
>and actually asked my physics GSI more than once, without a
>satisfactory answer.
>
>First, _where_ do we get uncertainty principle? The answer I got was
>the one I posted here: he said that it comes from cannonical
>commutator, [x,p] = ihbar (at least for the case of position and
>momentum.

Yes. Same for any pair of conjugate variables.

> I know, from general uncertainty theorem, that operators that
>commute have simultaneous eigenstates, thus can have exact values at
>the same time). But then, my next question was, where does [x,p] =
>ihbar come from? Now, that's something just assumed, he said (as can be
>seen in "cannonical" commutator). I was told that I could start at
>either end--either by assuming that cannonical commutator, [x,p] =
>ihbar, or by assuming the position-momentum uncertainty principle,
>reach the other end (and as you know, all the other operators...(er...
>with the exception of quantum spin... I'm not quite sure about that...)
>can be defined in terms of momentum and position, and we can get their
>commutator... and so on.
>
Well, in one formulation of QM (an early one, by Dirac) [x,p] = ihbar
is a postulate. But it is not coming out of thin air. By the time
Dirac was working on it, we already had an example of a physical
phenomenon which can be described using both wave theory (as an exact
represention) or particle mechanics (as an approximation). I'm talking
here about optics. The "ray description" which was the early one is
just standard particle mechanics. Seemingly quite different from wave
propagation yet one is obtained as the limit of the other. So dirac
could use this as a model. And what said transition from one to
another involves is a relationship between commutators in the wave
description and Poisson brackets (which are an important concept in
classical mechanics) in the ray one.

Now, the Poisson brackets of canonical variable have the property that
they are usually zero except when two conjugate variables (like
position and the corresponding momentum) are involved. In this case
they're non-zero (one, to be exact). So, Dirac was postulating a
similar relationship for the commutators (corresponding to Poisson
brackets) in QM.

In a later formulation of QM< by Feynman, the transition between
classical mechanics and QM is done through the Least Action Principle
by generalizing it, again in the same way it occurs between classical
mechanics and classical wave theory. IN this case, the rule regarding
commutators just follows from the LAP, it is not an independent
postulate.

>Second, about the difference between classical wave and QM... my GSI
>(the same one) told me that if you study enough of classical wave
>theory, it looks much like QM, that mathematically, there is not much
>difference. So, what makes classical wave deterministic and QM
>nondeterministic? Is it because in QM, the wave is not the kind of wave
>you see on surface of water, but a representation of probability
>distribution?
>
Well, think about what the classical wave is. The basic thing here is
not wave but a field. We've these two kinds of physical entities.
One one hand we've particles which are localized. On another, we've
fields which are physical entities defined over some region of space
(possibly all space), with a position (and possibly time) dependent
values over the whole region. Doesn't have to be anything exotic.
You can talk about the pressure field, in the atmosphere, or the
"water level field" on the surface of a lake, etc. And when we're
talking about a wave, we mean by this a field with a space and time
variation which satisfies a specific relationship (a wave equation).

Now, the thing about a field is that at the same time it is "here, and
there and everywhere". When we're talking about EM waves (with the
field quantity being the EM field) or water waves (with the field
quantity being the water level) the wave equation predicts the values
the field takes in all possible locations at any given time and all
these values are there.

In QM, though, you ran into a small problem. The dynamics of a
particle is represented by a wave. But the fundamentl property of
"particlehood" is that particles are localized in the sense that when
you perform a measurement you find them "here *or* there, certainly
not everywhere". So, we've a small problem. We've a wave equation
which, as I mentioned, is perfectly deterministic and which gives you,
for any point in time, the wave values in all spatial locations, but
whenever we ceck where the particle is we find it just in one
location. So the wave cannot directly represent the presence of a
particle. It can only do so indirectly. And out interpretation
(supported by experiment) is that the field the is waving is a
probability field.

So, to answer your question, yes, the nondeterminism is brought in by
the fact that the wave only provides a probability distribution. And
once you've probabilities you've no determinism.

>
>PS. Oh, so... you are saying that uncertainty principle is not the
>basis on which QM is indeterministic? I'm just curious--then why is QM
>(what is the principle, postulate, or axiom behind QM that makes it)
>indeterministic?
>
You said it yourself, above. The fact that the wave provides only
probabilities. Determinism is all about one-to-one relationships.
A given initial situation giving rise to a one and only one outcome.
"If A then B", that's deterministic, "if A then 70% chace of B and 30%
chance of C", that's not.

Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"

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