Re: Query about intrinsic verus orbital angular momentum

torre@xxxxxxxxxx wrote:
You say "one can show angular
momentum of the r x p type (associated with rotation) can have only
integer quantum numbers." My question simply, is, HOW does one show
this? Especially, how does one *exclude* spin 1/2 from also being of
the rxp type?



This comes from trying to define position, momentum,
and orbital angular momentum as self-adjoint operators.

A nice discussion of some of the issues is in

E. Merzbacher
"Single-valuedness of Wave Functions"
American Journal of Physics, Vol. 30, pgs. 237-247 (1962).

Other relevant papers are:

C. Van Winter, Annals of Physics 47, 232 (1968).
S.S. Sanikov, Nucl. Phys. 87, 834 (1967).

For an old discussion see
W. Pauli, Helv. Phys. Acta 12, 147 (1939).

Van Winter provides a very detailed discussion. He chooses
a set of spherical harmonics with half integer values of l,
and shows that orbital angular momentum operator is not
self-adjoint with respect to such a set of functions.
In the case of l = 1/2, his set of functions YW_{lm}
(up to a normalization constant which I omit here) is:

YW_{1/2,1/2} = sin^{1/2} \theta e^{i phi/2}
YW_{1/2,-1/2} = sin^{1/2} \theta e^{-i phi/2} (1)

But there is a problem with the above set of functions,
because the second function does not come from the first
by the application of the "ladder" operator L_. Namely,
L_ YW_{1/2,1/2} is not equal to YW_{1/2,-1/2}.
Instead, we have that L_ YW_{1/2,1/2} is proportional to
cos \theta sin^{-1/2} \theta e^{-i phi/2}.

Other authors, including E. Merzbacher, also use the above
set of functions (and its generalization for higher l and m
values), and consequently they do not obtain self adjoint
orbital angular momentum operator.

In this respect, it is more natural to take functions

Y_{1/2,1/2} = (i/\pi) sin^{1/2} \theta e^{i phi/2}
Y_{1/2,-1/2} = -(i/\pi) cos \theta sin^{-1/2} \theta e^{-i phi/2}

for which it holds L_ Y_{1/2,1/2} = Y_{1/2,-1/2}.
We also have:

L+ Y_{1/2,1/2} = 0
L+ Y_{1/2,-1/2} = Y_{1/2,1/2}

But there is a catch: The action on L_ on the function
with lowest m value, i.e., m = -1/2 in our example,
we find

L_ Y_{1/2,-1/2} = Y_{1/2,-3/2}

The result is not zero, as it should be. And yet, if we
act on the latter state with the raising operator L+,
we obtain zero:

L+ Y_{1/2,-3/2} = 0.

The last result is crucial. It has for a consequence,
that the operators L_x, L_y, L_z are self adjoint,
provided that one employs a suitable renormalization
of the inner products between certain states. Then
everything appears to be OK, and the spherical harmonics
with half-integer l-values do form an irreducible
representation of the 3-dimensional rotation group.

All this has been discussed in papers
D. Pandres, J. Math. Phys. 6, 1098 (1965),
D. Pandres and D.A. Jacobson, J. Math. Phys. 9, 1401 (1968),

I followed Pandres' approach in a paper of mine

"Rigid particle and its spin revisited"
Published in Found.Phys.37:40-79,2007.
http://arxiv.org/abs/hep-th/0412324
in which I tried to clarify some other aspects as well,
and also used a complementary functions Z_{lm} in
a superposition with Y_{lm}, in order to obtain
"good behavior" under rotations, parity and time reversal.
Pandres and I do not claim that the ordinary
orbital angular momentum can have half integer l-values.

A reason of why to reject Y_{lm} and Z_{lm}
with half-integer l-values in the description of orbital angular
momentum was given correctly by Dirac in his textbook on QM.
In the free case, a complete set of solutions to the
Schr\" odinger equation consists of plane waves,
which are single valued. The latter property has to be preserved
when we use another representation, i.e., one with spherical
harmonics.
Hence, such Schr\" odinger basis for spinor representation of the
3-dimensional rotation group cannot refer to the ordinary configuration
space of positions, but to an internal space associated with
every point of the ordinary space.

Double valued spherical harmonics are admissible,
if they do not refer to the ordinary configuration space in
which the usual quantum mechanical orbital angular momentum is defined,
but if they refer to an internal space in which a spin angular momentum
is defined. An example of such an internal space is the space of
velocities, associated with the so called {\it rigid particle}
whose action contains the square of the
extrinsic curvature of a particle's world line.

`Fermion' spherical harmonics are discussed also in papers:

G. Hunter, P. Ecimovic, I.M Walker, D. Beamish, S. Donev, M. Kowalski, A. Arslan
and S. Heck, J. Phys. A: Math. Gen. 32 795 (1999); G. Hunter and M. Emami-
Razavi, \Properties of Fermion Spherical Harmonics", [arXiv:quant-ph/0507006].

.

Relevant Pages

  • Re: Query about intrinsic verus orbital angular momentum
    ... how does one *exclude* spin 1/2 from also being ... and orbital angular momentum as self-adjoint operators. ... a set of spherical harmonics with half integer values of l, ... representation of the 3-dimensional rotation group. ...
    (sci.physics.research)
  • Re: Query about intrinsic verus orbital angular momentum
    ... how does one *exclude* spin 1/2 from also being ... and orbital angular momentum as self-adjoint operators. ... a set of spherical harmonics with half integer values of l, ... representation of the 3-dimensional rotation group. ...
    (sci.physics.research)