Re: quantum mechanics of dissipative systems

["John S. Denker" <jsd@MONMOUTH.COM>]

At 07:52 PM 12/17/99 +1100, Margaret Mazzolini wrote:
> I had thought that the standard popular science type discussion of
> Heisenberg's uncertainty principle, where it is claimed that there is a
> ridiculously small but non-zero probability that an elephant could
> quantum-tunnel through a wall, was falacious because the uncertainty
> principle doesn't apply to a macroscopic, incoherent assembly of
> particles like an elephant.

Some remarks:

1) For all practical purposes, elephants do not tunnel through walls.  This
is true with a wide safety margin.  Experience is consistent with
theory.  The theoretical conclusion is unchanged even if we make a number
of very crude approximations -- or even gross errors  -- in the
calculation.  Therefore this is not a very informative model system.

2) Vastly more insight into the real physics can be obtained by considering
ammonia, which is sufficiently small that it continually "tunnels" as
described in _The Feynman Lectures on Physics_  volume III chapter 9.  The
isotopically labelled version, NHDT, tunnels also, but at a greatly reduced
rate because of the higher masses.  Any attempt to draw a classical
stereochemical diagram of NHDT is manifestly unphysical, because the
classical diagram is necessarily chiral whereas the real molecule exists in
a state that is a superposition of the two chiral basis states.

We can contrast this with almost any molecule bigger than NHDT, for which
the classical diagram is just fine.  The fact that we think of small
molecules such as alanine as chiral tells us that the inversion process has
been strongly suppressed.  For really big molecules, mass and distance can
explain a really small tunneling rate, but even if we stipulate a small
tunneling rate we need to explain why we virtually always observe the
molecules in states of definite dipole moment rather than definite parity
or any of the innumerably many other points on the sphere that represents
the quantum-mechanical state space.

In general, the only way to reconcile the existence of chiral molecules
with the fundamental laws of quantum mechanics is to invoke
dissipation.  For molecules larger than NHDT, the environment interacts so
strongly with them that it tends to lock them into preferred states,
greatly suppressing tunneling.  Often it is the states of definite dipole
moment that are preferred, because it is the dipole moment that produces
dissipation by coupling to the environment.

3) The laws of quantum mechanics can be used to correctly describe
macroscopic systems including dissipative systems.  This includes
elephants, alanine, and everything else.

4) The uncertainty principle is a consequence of the laws of quantum
mechanics.  However, as a practical matter, it is not a good starting place
for calculating tunneling rates.  So it is best to talk about quantum
mechanics in general, not the uncertainty principle in particular.

5) Quantum-mechanics textbooks typically consider only the low-dissipation
limit.  The dissipative case is a bit trickier, but doable.  There are
innumerable ways to misstate the laws of physics.  So (to answer the
original question), typical popular-science discussions are fallacious
because they unjustifiably neglect the environmental interaction modes --
not because there is any known limit to the applicability of the
fundamental laws of quantum mechanics.