Rayleigh–Ritz method - Yukawa coulomb potential

[AwesomeTrains]

Hello everyone

Homework Statement

I have been given the testfunction \phi(\alpha, r)=\sqrt{(\frac{\alpha^3}{\pi})}exp(-\alpha r), and the potential V(r,\theta, \phi)=V(r)=-\frac{e^2}{r}exp(\frac{-r}{a})
Given that I have to write down the hamiltonian (in spherical coordinates I assume), and I have to calculate the angular momentum operator \hat{L}^2 \phi. (This is only a part of the whole problem. a) of a), b) and c) They should have used some other symbol for the testfunction than \phi, it's kinda confusing)

Homework Equations

Angular momentum operator in spherical coordinates.

The Attempt at a Solution

I guess the answer is 0, because \hat{L}^2 \phi contains derivations of \theta, \phi which the testfunction doesn't depend on. Is this true?

 

[blue_leaf77]

Hello everyone

I guess the answer is 0, because \hat{L}^2 \phi contains derivations of \theta, \phi which the testfunction doesn't depend on. Is this true?

Yes that's true. Another way to look at it is to realize that the test function is proportional to $Y_0^0$.