Re: angular momentum raising lowering ladder operators

corpsicle wrote:
hello,

are there any angular momentum operators for, say, a hydrogen atom,
that raise and lower the angular quantum number l (ell) up and down,
as opposed to the Lx + iLy type operators that move the magnetic
quantum number m, "sideways" as it were but leave l unchanged?
so what i guess i'm asking is there some kind of differential
equation that generates the l+1'th spherical harmonic from the l'th,
or, or, something!

If I may, your question can be rephrased as: is there an identity
relating spherical harmonics Y_lm(theta,phi) with different l's but
fixed m, theta, and phi? After all, the Hilbert space of any system
with rotational degrees of freedom can always be factored into a tensor
product functions on the 2-sphere and functions depending on other
degrees of freedom.

The answer is yes, and it need not even involve differentiation. Up to
coefficients, each spherical harmonic is of the form[1]

Y_lm(theta,phi) ~ exp(i m phi) P_lm[cos(theta)] ,

where P_lm(z) is an associated Legendre polynomial. There is a
collection of identities, recurrence relations and other formulas for
associated Legendre polynomials at Wolfram Research's database of
special functions[2]. There are likely a few more formulas found in the
venerable Handbook by Abramowitz & Stegun.

You can adapt any one of these identities to Y_lm as well. I don't know
if any of these recurrence relations will turn out to be particularly
nice. If you find one that is, let us know!

Hope this helps.

Igor

[1] http://functions.wolfram.com/Polynomials/SphericalHarmonicY/02/
[2] http://functions.wolfram.com/Polynomials/LegendreP2/17/01/01/

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