I've been trying to prove a rather simple looking concept. I have a code that calculates states of a 3D anisotropic oscillator in spherical coordinates. The spherical harmonics basis used to expand it's solutions in radial coordinate constraint the spectrum such that when the Hamiltonian is diagonalized it calculates only states with L(adsbygoogle = window.adsbygoogle || []).push({}); _{z}=0, because the potential has a spherical harmonic (Y_{1}^{0})^{2}. i.e.

V_{HO}=1/2[itex]\hbar[/itex]m [[itex]\omega[/itex]_{xy}^{2}(x^{2}+y^{2}) + [itex]\omega[/itex]_{z}^{2}z^{2}]

V_{HO}=1/2[itex]\hbar[/itex]m [[itex]\omega[/itex]_{xy}^{2}(x^{2}+y^{2}+z^{2}) + ([itex]\omega[/itex]_{z}^{2}-[itex]\omega[/itex]_{xy}^{2})z^{2}]

Since z=rCos(θ)

V_{HO}=1/2[itex]\hbar[/itex]m [r^{2}([itex]\omega[/itex]_{xy}^{2}+ ([itex]\omega[/itex]_{z}^{2}-[itex]\omega[/itex]_{xy}^{2})2[itex]\pi[/itex]/3(Y_{1}^{0})^{2}]

Now, we know the system spectrum in Cartesian E_{Nx,Ny,Nz}= 1/2[itex]\hbar[/itex][[itex]\omega[/itex]_{xy}(N_{x}+N_{y}+1) + [itex]\omega[/itex]_{z}(N_{z}+1/2)]. So to calculate this spectrum on paper for verification one can either

(a) calculate spectrum for 3D anisotropic oscillator in spherical coordinates directly OR

(b) look for states with L_{z}=0 in terms of N_{x}, N_{y}, N_{z}by introducing constraints on N_{x}, N_{y}& N_{z}-> like N_{x}=N_{y}, N_{z}=0 OR N_{x}=2N_{y}, N_{z}always even or some such rules..

Does anyone have advice on how to derive the anisotropic oscillator spectrum in spherical coordinates (using regular spherical harmonics Y^{m}_{l}as basis for solutions).. if not any advice on how to derive constraints on N_{x}, N_{y}, N_{z}to give L_{z}=0 states only?? Any help will be greatly appreciated.. thanks!!

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# 3DAnisotropic oscillator in Spherical Harmonic basis-States with L_z=0

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