From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: quantization of angular momentum
Organization: Computer Science Department,  Stanford University.
Date: Mon, 7 Sep 1992 21:55:11 GMT

In article <25949@dog.ee.lbl.gov> sichase@csa3.lbl.gov writes:
>>>>Ouch!  L is quantized under all circumstances.  Its eigenvalues are
>>>>half-integer multiples of hbar, even for free particles.
>>Angular momentum, though, always has discrete eigenvalues.
>
>Angular momentum is not a good quantum number for a plane wave.

I don't agree.  Plane waves and the trajectory of the particle are
surely a red herring here.  Start with the classical picture.  The
orbital angular momentum of a particle with position r and momentum p
is defined to be r x p, a vector normal to both r and p.  In this
definition there is no restriction on the shape of the trajectory, and
no requirement that the origin (with respect to which r is measured) be
the center of curvature of the trajectory.

To better see this (still in the classical framework), consider a stone
whirled around at one end of a piece of string whose other end is held
stationary at a point we take to be the origin.  When you cut the
string what happens to the orbital angular momentum of the stone?
Answer: it is conserved.  Since the stone is now traveling in a
straight line (ignoring gravity) this would appear to contradict
Scott's intuition about its orbital angular momentum.  But when you
consider the definition of angular momentum, rxp, this conservation can
be seen to be an immediate consequence of this definition and the
choice of center of rotation for the origin, a choice we did not change
when we cut the string, even though the stone no longer describes a
circular orbit about that origin.  The stone maintains a fixed angular
momentum about *our choice of origin* as it zooms away on its
now-linear trajectory.

Now pass to the quantum picture.  All of the above remains true, except
for the outcome of measurements of the above quantities.  Measurements
of the orbital angular momentum now yield integer multiples of h~
(writing h~ for h-bar --- note that half-integers occur for spin AM but
not for orbital AM).  Measurements of either one of r or p can be made
to arbitrary precision, but not both together, by Heisenberg (which I
think is also a reasonable answer to Tom Knight's original question).

Since Tom put in a request for additional intuition, here's a picture I
have found helpful in visualizing the quantum mechanics of L = r x p,
and more generally of any conjugate pair of observables.  If we take r
and p to be points in the n-dimensional spaces R and P, then we can
view the pair (r,p) as a point in the 2n-dimensional space RxP.  (Take
n=1 to visualize this, n=3 to apply it to the 3D world, necessary to
make sense of angular momentum.  Here RxP is direct product of vector
spaces, not to be confused with the vector product rxp which assumes
n=3 and is a bilinear transformation mapping the 6D space RxP to the 3D
space of angular momenta.)

In classical mechanics RxP is called *phase space*.  Phase space
behaves like an incompressible fluid: as time passes, regions in phase
space may change shape but not volume.

Now in quantum mechanics, points in phase space have an uncertainty of
h~, that is, points blur into regions of volume h~.  Now if we bring
the classical incompressibility of phase space into the quantum picture
then we find that a region of uncertainty retains its overall volume.
But its shape may change: it can become arbitrarily narrow in either
its position or momentum component.  This corresponds to knowing
respectively the position or momentum of the particle to arbitrary
precision.  But we cannot know both to arbitrary precision since that
would require the volume of the region of uncertainty to decrease below
h~.

This picture applies anywhere that Heisenberg's principle applies.  Its
key ingredients are the classical incompressibility of phase space
applied to quantum mechanics, and the "amorphously quantized"
uncertainty of points in quantum phase space.  This gives a graphic way
of visualizing Heisenberg's uncertainty principle in general, and in
particular the paradox of quantized L = r x p in the face of continuous
r and p.

What I don't know is how robust this picture is, i.e. where if anywhere
does it let one down?
-- 
======================================================| God found the positive
Vaughan Pratt   pratt@cs.Stanford.EDU   415-494-2545  | integers, zero was
======================================================| there when He arrived.
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