Eigen values for a state and spherical harmonics

[samee]

Homework Statement

The complete wavefunction for a particular state an atom, is Si(r,theta,phi)=Ne^(-Zr/a_0)(Z/a_0)^3/2sqrt(1/4pi). What is the eigenvalue Lz for this state?

Homework Equations

see above

The Attempt at a Solution

This is the last one that I'm having trouble with. I have no idea how to start it. Just some pointers on how to begin would be awesome...

 

[vela]

The hydrogenic wavefunctions are of the form \psi(r,\phi,\theta)=R_{nl}Y_l^m. You've been given the wavefunction
\psi = N r^2 e^{-\frac{Zr}{3a_0}}\sin^2 \theta e^{2i\phi}.

Identify the pieces and see if you can identify what n, l, and m are for this state.

 

[samee]

Okay, so since R_n,l=2(Z/a₀)^3/2 e^-Zr/a₀, it means that the first part of the wavefunction is R and the second part is Y?

I know that Y_0,0=Sqrt(1/4pi)
which fits the equation except for the constant out front.

This means that l=0. The first R for which l=0 is R_1,0=2* e^(-Zr/a₀)(Z/a₀)^3/2

So, if N=1/162sqrt(pi) *(Z/a₀)⁷, then this wavefunction is R_1,0Y_0,0

This means that it is in the state |1,0,0>

 

[vela]

Good work. Now learn how the quantum numbers n, l, and m relate to Lz. (You should also know how they relate to other observables, like the energy and total angular momentum, though you don't need to know that for this particular problem.)