Angular momentum of a particle in a spherically symmetric potential

[stunner5000pt]

Homework Statement

A particle in a spherically symmetric potential is in a state described by the wavepacked

$$\psi (x,y,z) = C (xy+yz+zx)e^{-alpha r^2}$$

What is the probability that a measurement of the square of the angular mometum yields zero?
What is the probability that it yields [tex] 6\hbar^2 [/itex]?
If the value of l is found to be 2. what are the relative probabilities of m=-2,-1,0,1,2

2. The attempt at a solution

i think the first part is simply aking to calculate $<L^2>$

but the carteisna coords are throwing me off... Should i convert to spherical polars?? Till now whenever the angular momentum L^2 and Lz were required, they were gotten using
$$\hat{L^2} \psi_{nlm_{l}} = l(l+1) \psi_{nlm_{l}}$$

really from the spherical harmonics... however conversion to spherical polars doesn't yield any familiar spherical harmonic either.

can it written in a way that yields familiar spherical harmonics, however??

 

[Dick]

Yes. Convert to spherical coordinates. You won't necessarily get a spherical harmonic but you can decompose it into spherical harmonics in the usual way you split a wavefunction relative to an orthonormal basis.