From: mcirvin@husc8.harvard.edu (Mcirvin)
Newsgroups: sci.physics
Subject: Angular momentum thought experiment
Date: 3 Sep 92 01:39:31 GMT
Article-I.D.: husc3.1992Sep2.213933.15325
Distribution: world
Organization: Harvard University Science Center

A while back someone posted a question about measuring different
components of the angular momentum of correlated particles, in a
setup where everything is bolted to a rigid chassis.  The question
was, I believe, whether non-integer or even arbitrarily large
values of angular momentum for the system (or for the photons) can
occur as a result of doing this repeatedly.  Whoever it was
(I've totally forgotten) is probably rather incensed by now at
the total lack of response, which was, incidentally, due to the
fact that it's a somewhat tricky problem.  I shall try to remedy
the situation by attempting an answer.

The basic situation described was equivalent to what I'm going
to describe.  The question originally involved photons and 
polarizers, but I'm going to talk about spin 1/2 particles and
idealized "Stern-Gerlach filters" of the type used by Feynman
in his thought experiments, since it's easier for me to think 
about them.

Such a filter is capable of selecting out particles whose
eigenvalue of spin in a certain direction is +1/2.  Thus they
are the equivalent of polarizers.  Suppose we have a source
in the middle of the rigid chassis which emits pairs of 
fermions whose spins are correlated in a certain way; for
concreteness let's say they're in a singlet state, where
in any direction the state is proportional to
|up> |dn> - |dn> |up>  (properly normalized, of course).

The two particles travel in opposite directions and pass
through the filters.  If the filters are oriented in the
same direction the particles will always be anticorrelated:
if one gets through one filter, the other will be blocked.
If the filters are oriented in opposite directions the
particles will either both get through or not get through
at all.

If they both get through, it's clear that the angular momentum
hasn't changed.  If one is blocked, that particle was absorbed
by the filter, so it's clear what happened to the angular
momentum.

But what if we separate the filters by some other angle?
Suppose that they're oriented in directions separated by
90 degrees.  Then if particle a gets through filter a,
particle b has a 50% chance of getting through filter b.
Suppose it does.  What is the angular momentum state now?
How much angular momentum is there?

The important thing to remember is that you can't just
designate an angular momentum state in quantum mechanics
by a vector in three-dimensional space.  The components of
angular momentum are non-commuting observables, and have
uncertainty relations between them, just like position and
momentum.  So if we've measured particle a to have spin +1/2
(imagine the factors of hbar-- I'm a particle guy, and I
like to set hbar=1) in the z direction, and particle b to
have spin +1/2 in the x direction, we can't just add the
vectors and say that the two particles are in an eigenstate
with an eigenvalue of +1/sqrt(2) in a direction tilted at
45 degrees.  That is, in the words of Leader Kibo, Not
Allowed.  The system of two particles can only be in a
spin 1 or spin 0 state, so its eigenvalue of spin in any
direction can only be 1, 0, or -1.  (It started out in a
spin 0 state, the way I defined it.)

What do we do?  We examine the actual quantum state.
This is a vector, not in real space, but in Hilbert space:
the space of possible states.  We shall pick as basis
vectors in Hilbert space the eigenstates of spin in the
z direction for the two particles.  They are |up>|up>,
|up>|dn>, |dn>|up>, and |dn>|dn>, where |up> means
spin up and |dn> means spin down.  The state starts out
as 1/sqrt(2) (|up>|dn> - |dn>|up>).  Let's say that the
passage through the filter a happens first (it doesn't
matter what our ordering is, since-- let's all repeat it
again-- Spacelike Observables Commute!) Filter a is oriented
to let through spin-up particles, so if it gets through the
state becomes |up>|dn>.

Notice that we've already had a possible transfer of
angular momentum, because |up>|dn> is a superposition of
states with overall angular momentum 0 and overall angular
momentum 1.  By conservation of angular momentum, if nothing
gets emitted that can carry off the spin (we are being *very*
idealized here), the rigid chassis is in a superposition of
angular momentum eigenstates too!  Of course, they differ by a
really, really tiny amount.

Now particle b goes through filter b.  This filter is
oriented at a 90-degree angle to the first filter (the
logical equivalent of 45-degree angles for polarizers!),
so that it lets through particles with spin +1/2 in the
x direction.  Let's designate the x spin eigenstates
as |+x> and |-x>.  It so happens that |dn> is
1/sqrt(2) (|+x> - |-x>).  So in the new basis where we
refer to x eigenstates for particle b, the state coming
into the filter is |up> (1/sqrt(2)) (|+x> - |-x>).
If particle b gets through filter b, the state is now
|up> |+x>.  In terms of z eigenstates, this is
1/sqrt(2) (|up>|dn> + |up>|up>).  In terms of x
eigenstates, it's 1/sqrt(2) (|+x>|+x> + |-x>|+x>).
Notice that along either axis, we have a superposition
of a state in which the total angular momentum in that direction
is 1 and in which it is 0.  The same is true along *any*
other axis, but the amplitudes for the various possibilities
vary.  The rigid chassis has been thrown into a
superposition as well, so that in every case there is
no net change in angular momentum.

Now, you could also measure the *expectation value* of
the total angular momentum of the particles, the value you'd
get on average from doing a large number of experiments.  That
would be some number between 0 and 1.  It would never be
a number greater than 1, because the only possibilities for
the system of two particles of spin 1/2 are zero and one.*
 
I've probably screwed up mathematically somewhere, but
this is the essential idea.  An eigenstate of one
component of angular momentum is not necessarily an
eigenstate of another.

Looking at spins is an excellent way to get to the true
flavor of quantum mechanics.  


* This is, of course, neglecting the ideas of Yakir Aharonov
concerning pre- and post- selection, which I don't understand
anyway.  I think he claims that it is sometimes useful to
think in terms of values outside the range that one can get
by an ordinary weighted average of eigenstates.  In the
Copenhagen-like sort of picture that I've sketched here, these
considerations don't arise. 
-- 
Matt McIrvin, Cambridge, Massachusetts, USA 
Newsgroups: sci.physics