Homework Help: Eigen values for a state and spherical harmonics

1. Feb 23, 2010

samee

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

see above

3. 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...

Last edited: Feb 23, 2010
2. Feb 23, 2010

vela

Staff Emeritus
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.

3. Feb 23, 2010

samee

Okay, so since Rn,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 Y0,0=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 R1,0=2* e^(-Zr/a0)(Z/a0)3/2

So, if N=1/162sqrt(pi) *(Z/a0)7, then this wavefunction is R1,0Y0,0

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

Last edited: Feb 23, 2010
4. Feb 23, 2010

vela

Staff Emeritus
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.)