Quantum-Classical limit of Zonal Harmonics

["pi"mp]

Hi all, I'm studying the spherical harmonics, specifically the zonals (set m=0). With the equatorial harmonics(set m=l), the classical limit is pretty darn clear; the particle just accumulates to a Dirac delta along the equator, but I've proven that the zonals don't converge to a Dirac Delta at the north and south pole. They certainly accumulate there, but they converge to a fixed Bessel function instead of a genuine Dirac. Does anyone have any ideas on the classical picture emerging here? I've heard some talk of quantum scattering but I'm really not sure.

My personal thought is that with the equatorials, there is really only one plane perpendicular to the z axis and we know the angular momentum vector converges to the z-axis ==> by symmetry, they must necessarily be "orbiting" in a plane in the limit. However, with the zonals, all we know is that the z component of the angular momentum is zero. But there are infinitely many planes with normals vectors of zero z component. Is there some kind up superposition going on here that's preventing us from getting a genuine Dirac?

Thanks!

 

[Jano L.]

With the equatorial harmonics(set m=l), the classical limit is pretty darn clear; the particle just accumulates to a Dirac delta along the equator, but I've proven that the zonals don't converge to a Dirac Delta at the north and south pole.

What do you mean by classical limit - limit of one Y function for large [itex]l[/itex]?

And what is mathematical meaning of "particle just accumulates to a Dirac delta" ? That the probability density [itex]|\psi|^2[/itex] localizes around the equator and falls to zero elsewhere?

There is no reason why the spherical functions should give localized prob. distribution around poles or elsewhere - typically these functions are non-zero all over the sphere.

 

["pi"mp]

I've proven that the modulus square of the equatorial harmonics (m=l) genuinely converge to a "line Dirac Delta" so to speak at the equator. This is the classical limit because it corresponds to orbital motion (in the limit, of course)

However, the zonal harmonics (m=0) accumulate at the north and south pole but they're not quite genuine Dirac Delta functions. The zonals converge to a Bessel functions with a slightly modified argument. Buy yes, in all of this I mean l going to infinity.

I was just wondering if anyone knew why the zonals converge slower.