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 $Y_0^0$.