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Re: [Phys-l] distinguishability

On 07/16/2011 11:42 AM, Carl Mungan wrote:
Regarding question 2, I see that I didn't word it properly. I'll try
again:

2. Is there such a thing as an ideal gas of identical but
distinguishable quantum particles?

As a specific example, suppose we are talking about one million atoms
of helium-4 and all atoms have their electrons filling the 1s
orbital. Thus all atoms are identical. But put these atoms in a
cubical box (that is otherwise empty) measuring 100 meters on a side.
It seems like each atom will have on average a cubic meter to itself,
so I can distinguish atoms (at least for some time) by saying there's
one in that cubic meter, one in that cubic meter, etc.

To put question 2 another way, does anything special happen when D
L > R where L is the thermal de Broglie length, R is the size of
a gas particle, and D is the average distance between gas particles?
I put >>> to mean hugely bigger, not just the ordinary bigness of
STP. Special might mean the Sackur-Tetrode equation changes, for
example.

That's a perfectly well-posed question. The short answer is yes,
you can arrange for particles to be distinguished by their positions
... for a while. And yes, this changes the Sackur-Tetrode equation
and everything else.

That answers the question that was explicitly asked. So far so good.

Now all that remains are about 100 follow-up questions. This includes
seemingly-minor technical questions, such as how to set up such a state,
how long it will last, et cetera. But at a more profound level, I consider
question 2 to be a thinly-disguised way of asking:

3. How do we identify -- and understand -- the boundary between the
quantum-mechanical world and the familiar classical approximation?

You could hardly ask a deeper question.

This question actually does have an answer, but to fully explain it
would take at least 100 pages. It requires understanding the foundations
of quantum mechanics and the foundations of thermodynamics. It is
tangentially related to a lot of other famous questions, including
quantum computing, Schrödinger's cat, and quantum measurement theory.
Sub-questions include:

3a) Why is an ammonia molecule different from almost any other
molecule? Why is the classical ball-and-stick model wrong for
a free ammonia molecule, but OK for alanine and all larger
molecules, and even OK for ammonia in solution? What do you
have to do to ammonia to make it classical? Where is the
boundary? Why?

3b) In a SQUID, under what conditions do you see macroscopic quantum
tunneling, or not? Why?

Here are some hints:

A) Read Feynman and Hibbs, _Quantum Mechanics and Path Integrals_
especially the part about thermodynamics.

This is not what I would call light reading.

Also beware that there are scads of misprints ... but the physical
concepts are right, and "most" of the equations are right.

B) There are three different concepts:
-- identical particles
-- indistinguishable situations
-- exchange

You really need all three. The idea of _exchange_ is the most
advanced, because (unlike the others, which are yes/no questions)
you get to talk about the _amount_ of exchange, i.e. the probability
of exchange. If the probability of exchange is small, it doesn't
much matter whether the particles are identical or not.

The next few paragraphs elaborate on this.

C) If you put each helium atom in a box with impenetrable walls, you
know there will be no exchange. The effect of the walls can be
quantified in terms of a large potential energy, which is a barrier
to exchange.

As a more practical example, the D-alanine molecule is separated
from the L-alanine molecule by a potential barrier (in reaction-
coordinate space) that plays a key role in preventing tunneling
from one configuration to the other. The potential barrier is
important, but it is not the whole story; see items (E) and
especially (F) below.

D) More relevant to question (2) and question (3) is the fact that
there is also an imaginary-kinetic-energy contribution that affects
the probability of exchange. In the path-integral formulation,
thermodynamics can be considered the analytic continuation of
quantum mechanics. The inverse temperature (beta) is in some
sense the imaginary part of the time.

The method of stationary phase (with respect to time) becomes the
method of steepest descent (with respect to beta).

Specifically: The atoms get to play musical chairs. Imagine turning
out the lights and waiting for a time βℏ, during which time atoms
can switch places, if they can do so and return to an equivalent
state before the time is up. And if the distances are too large
and/or the time is too small, there is a kinetic-energy term in the
equation of motion that reduces the probability of exchange, as a
companion to the potential-energy term mentioned in paragraph (C).

This means that only atoms that are cold and/or quite close together
will undergo exchange with any appreciable probability.

Here's another way to say it: Whereas identical particles and symmetry
of the wavefunction are mathematical ideas, exchange is a _physical_
process. Exchange happens by means of real physical processes, and
can be blocked. (In contrast, symmetry cannot be blocked. Either
the wavefunction is symmetrical or it's not.)

This is super-important. Without it the classical limit would
not exist. Also, without it quantum mechanics would be completely
impractical, because we would never be able to talk about the wave-
function of an individual electron; we would have to antisymmetrize
the wavefunction with respect to every electron in the universe.

This is relevant to the question of how long the particles will remain
distinguishable by their position ... but there is more to the story.

E) If you idealize the system as closed, you will get wrong answers
to questions about the transition to classical behavior. That's
because the QM equations of motion for the closed system are unitary,
and 100% reversible. So if you know the initial state, you know the
state for all time, and the rate-of-change of entropy is zero.
Reversible evolution is a definite possibility, as we see in spin-echo
experiments, where something that is complicated and /seems/ entropic
can nevertheless be reversed; the movie can be played backwards ...
within limits.

On the other side of the same coin, the limits must be taken into
account. Any real-world system couples to the outside world at least
a little bit. The particles excite phonons when they hit the walls.
They also couple to the electromagnetic field. To the extent that
these excitations depart the system and are never seen again, they
represent an irreversible increase in entropy. For a system with
only a million atoms, its maximum-entropy state doesn't have a whole
lot of entropy anyway, so seemingly-small couplings make a significant
difference.

F) The coupling to the environment changes the statistics (i.e. the
entropy) ... and it also changes the dynamics, sometimes by a factor
of infinity. For more about the role played by the environment --
including "superselection rules" -- in causing the transition to what
we call classical behavior (which is of course still governed by the
laws of quantum mechanics), see:
Wojciech H. Zurek
"Decoherence, einselection, and the quantum origins of the classical"
Rev. Mod. Phys. 75, 715 (2003)
http://arxiv.org/abs/quant-ph/0105127

*) etc. etc. etc.