Re: Chirality, Copenhagen and Collapse

[John Denker <jsd@AV8N.COM>]

On netnews, zigoteau wrote:
> neither enantiomer is a true eigenstate of the system.

This is a deep, important question.  It's amazing how rarely
the question comes up, given its importance.  It is even
more rarely that a proper answer comes up.

As we shall see, this question illuminates (indeed practically
defines) what we mean by the border between quantum mechanics
and classical mechanics.

If you google for Hund's paradox
  http://www.google.com/search?q=hund%27s-paradox
you get virtually no useful information.  Apparently Prof.
Gustav Obermair at Regensburg has assigned this as a PhD
thesis topic to a couple of students, but neither figured
out the right answer.

> A well-documented example of a similar
> idea is ammonia, NH3, as used in the first masers.

Quite so.
A convenient reference is _The Feynman Lectures on Physics_
volume III chapter 9.

> Exactly the same situation must be the case for the l- and
> d-enantiomers of lactic acid, or potassium tartrate, or any other pair
> of chiral compounds.

Or so you would think, based on naive theory.  But the
facts are otherwise.  There are lots of honest-to-goodness
chiral molecules.

So the agenda is to come up with a theory that explains why
what's true for ammonia and a few closely-related molecules
is not true for the remaining vast majority of molecules.

> This appears to be in contradiction to the tenet of the Copenhagen

Nope.  It's got nothing to do with Copenhagen.  It's much
more fundamental than that.

> Of course the problem goes away on the Everett

Nope.  Everett solves nothing ...  As a rule, nothing in
general, and certainly nothing in this case.

===============

Let me explain why the facts raise even more questions than
have been mentioned so far.

Choose as basis states the two classical states, i.e. states
of definite parity, each a mirror image of the other.  By
symmetry, they have the same energy.  In this basis, there
must be some off-diagonal matrix element that leads to
interconversion.  In the case of ammonia, we know what
this matrix element is, and we observe the interconversion.
In the case of something heavier, such as alanine, we
imagine the matrix element is smaller, but still nonzero.

 -- The eigenstates of the hamiltonian will be the gerade
and ungerade combinations of these basis states.  We can
represent these as the north pole and south pole of the
SU(2) 'sphere'.  (It's not exactly a sphere, but close
enough for the present representational purposes.)

 -- The off-diagonal matrix element means that neither of
the basis states is an eigenstate of the hamiltonian.  They
will be nonstationary.  We can represent them as two
diametrically-opposite points on the equator of the SU(2)
'sphere'.  They rotate, marching around the 'sphere' at a
rate determined by the magnitude of the off-diagonal matrix
element.

For alanine, naive theory says this rotation will be much
slower than it is for ammonia ... but still nonzero.

This picture leaves us with *two* questions:
 a) The obvious question is: what makes the alanine states
stop rotating?
 b) The less-obvious question is: when they stopped, why
did they stop at the pair of points representing good-parity
states, as opposed to the uncountably many other points on
the equator where they could have stopped?

Question (b) is important, because it means that just saying
that the rotation is very slow does *not* solve the mystery.
A whole lot of non-experts just convince themselves that
the rotation is "slow" and imagine that they have solved the
problem, but they really, really haven't.

========

If you have an alanine molecule in a solution, it cannot flip
from left-handed to right-handed without re-arranging the
local solvent molecules.  So the mass involved is not just
the mass of the alanine, but also the mass of some of the
solvent.  The added mass further slows the rotation, but
there's more to the story, as we shall see in a moment.

If you have an alanine molecule in the vapor phase, it cannot
flip from left-handed to right-handed without re-arranging
the local electromagnetic field.

The added mass addresses question (a) above, but the
interesting part is that the environment (i.e. solvent
and/or EM field) is essentially _measuring_ what state the
molecule is in.

By way of analogy, suppose you have polarized light (initially
X-polarized) traveling through an optically-active medium.  The
polarization vector will rotate as it moves through the medium.
Now suppose that in the medium, very near the entrance, you
insert an X-polarizer.  This will project off the small Y
component that has accumulated, and restore the wave to pure
X polarization.  A short distance later, you do the same
trick.  And so forth.  In this way, you can stop the beam from
rotating by _measuring it to death_.  This is sometimes called
the "watched pot effect" (from the proverb that a watched pot
never boils).  I get 69 hits from
  http://www.google.com/search?q=quantum+watched-pot-effect

So this is what really makes classical states classical: the
watched pot effect.  Note that this answers question (b).
Just as the X polarizer locks the beam in the X state, the
EM field locks alanine in states of definite parity, because
the field couples to the molecule via its dipole moment.
Ditto for solvents.  In general, figure out which basis-states
couple to the relevant environment, and that tells you which
are the classical states.

Also, it turns out that the physics of measurement is intimately
related to the physics of dissipation, for reasons that I don't
feel like explaining at the moment.  So another way of expressing
this result is to say that where something sits relative to the
quantum / classical boundary depends on the amount of dissipation.
The cut-off is about 2Kohms ... that's hbar in the appropriate
units.  In particular, if you could suspend an alanine molecule
in a small shiny (i.e. superconducting) box so that it suffered
no radiative dissipation, and no other coupling to the environment,
it would do the SU(2) rotation just the way ammonia does (albeit
slower).