- #1
karlhoffman_76
- 5
- 1
See the picture below for details. I have included my solution thus far, but it doesn't feel right and I'm just wondering what I am missing. Any assistance would be greatly appreciated, cheers.
blue_leaf77 said:What about analyzing which powers of x,y, and z appear in the wavefunction and compare it with the spherical harmonics?
Comment to your second equation in your solution: ##\psi(\mathbf{r})## is supposed to be superposition of several eigenfunctions of ##L_z##, means you can't write the equation such that where you can associate the wavefunction with one value of m.
blue_leaf77 said:The purpose of the question is to check which eigenfunctions of ##L^2## and ##L_z## may appear when the original wavefunction is expanded as a sum of those eigenfunctions, which are the spherical harmonics ##Y_{lm}##. Now if you remember the characteristic dependency of ##Y_{lm}## w.r.t to the three coordinates x,y, and z is governed by the indices ##l## and ##m##. Using this fact you can get the first rough idea of which ##l## and ##m## are allowed to appear because they conform with the dependency of the original wavefunction to three coordinates. First of all, do you know how the spherical harmonics looks like and its dependency on x,y,z?
For example, ##Y_{2,-2} \propto (x-iy)^2##, by comparing this with the manner x,y, and z appear in the original wavefunction, do you think ##Y_{2,-2}## will appear in the expansion?
blue_leaf77 said:yes, that's what I think the correct one.
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