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srihari83
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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 Lz=0, because the potential has a spherical harmonic (Y10)2. i.e.
VHO=1/2[itex]\hbar[/itex]m [[itex]\omega[/itex]xy2(x2+y2) + [itex]\omega[/itex]z2z2]
VHO=1/2[itex]\hbar[/itex]m [[itex]\omega[/itex]xy2(x2+y2+z2) + ([itex]\omega[/itex]z2-[itex]\omega[/itex]xy2)z2]
Since z=rCos(θ)
VHO=1/2[itex]\hbar[/itex]m [r2([itex]\omega[/itex]xy2 + ([itex]\omega[/itex]z2-[itex]\omega[/itex]xy2)2[itex]\pi[/itex]/3(Y10)2]
Now, we know the system spectrum in Cartesian ENx,Ny,Nz = 1/2[itex]\hbar[/itex][[itex]\omega[/itex]xy(Nx+Ny+1) + [itex]\omega[/itex]z(Nz+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 Lz=0 in terms of Nx, Ny, Nz by introducing constraints on Nx, Ny & Nz -> like Nx=Ny, Nz=0 OR Nx=2Ny, Nz 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 Ymlas basis for solutions).. if not any advice on how to derive constraints on Nx, Ny, Nz to give Lz=0 states only?? Any help will be greatly appreciated.. thanks!
VHO=1/2[itex]\hbar[/itex]m [[itex]\omega[/itex]xy2(x2+y2) + [itex]\omega[/itex]z2z2]
VHO=1/2[itex]\hbar[/itex]m [[itex]\omega[/itex]xy2(x2+y2+z2) + ([itex]\omega[/itex]z2-[itex]\omega[/itex]xy2)z2]
Since z=rCos(θ)
VHO=1/2[itex]\hbar[/itex]m [r2([itex]\omega[/itex]xy2 + ([itex]\omega[/itex]z2-[itex]\omega[/itex]xy2)2[itex]\pi[/itex]/3(Y10)2]
Now, we know the system spectrum in Cartesian ENx,Ny,Nz = 1/2[itex]\hbar[/itex][[itex]\omega[/itex]xy(Nx+Ny+1) + [itex]\omega[/itex]z(Nz+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 Lz=0 in terms of Nx, Ny, Nz by introducing constraints on Nx, Ny & Nz -> like Nx=Ny, Nz=0 OR Nx=2Ny, Nz 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 Ymlas basis for solutions).. if not any advice on how to derive constraints on Nx, Ny, Nz to give Lz=0 states only?? Any help will be greatly appreciated.. thanks!
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