angular momentum

From: Tobin Fricke (fricke_at_ocf.berkeley.edu)
Date: 03/22/05 

Date: Tue, 22 Mar 2005 08:24:08 +0000 (UTC)

I would like to solidify my understanding of some very basic elements of
the theory of angular momentum in quantum mechanics.

An early step is always to define raising and lowering operators, e.g. the
raising operator J+ = Jx + i Jy. Clearly this works, and might be found
by playing around with operators until the raising/lowering operators were
"discovered." But it seems that operators involving X + i Y are somehow
'special'. For instance, x + iy can be thought of as a 'spherical basis
vector'. Is there a greater theory behind the raising/lowering operators?
How were they originally obtained?

The magic of the raising/lowering operators seems to come from the
commutation relations [Jz, J+/-] = +/- hbar J+/-. The harmonic
oscillator's operators have commutation relations of the same form. I
suppose any operators with a commutation relation like [X, Y] = c Y give
rise to some kind of raising/lowering operators for the eigenvalue of X ?

thanks,
Tobin