Rayleigh–Ritz method - Yukawa coulomb potential

  • #1
Hello everyone

Homework Statement


I have been given the testfunction [itex] \phi(\alpha, r)=\sqrt{(\frac{\alpha^3}{\pi})}exp(-\alpha r) [/itex], and the potential [itex] V(r,\theta, \phi)=V(r)=-\frac{e^2}{r}exp(\frac{-r}{a}) [/itex]
Given that I have to write down the hamiltonian (in spherical coordinates I assume), and I have to calculate the angular momentum operator [itex] \hat{L}^2 \phi [/itex]. (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 [itex]\phi[/itex], it's kinda confusing)

Homework Equations


Angular momentum operator in spherical coordinates.

The Attempt at a Solution


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

  • #2
blue_leaf77
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Hello everyone

I guess the answer is 0, because [itex] \hat{L}^2 \phi [/itex] contains derivations of [itex]\theta, \phi[/itex] 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##.
 

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