For the spherical case the wavefunctions live on the surface of a unit sphere. Because the interaction is not depending on the azimuthal angle φ, the (m-dependent!) effective Schrödinger equation of the generalized planar pendulum is solved for the polar angle 0 ≤ θ ≤ π, and the solutions are then multiplied with a phase factor exp(imφ). Exact solutions are obtained for k ≡ η / ( 2 √ ζ ) being integer.
First analytic solutions for the time-independent Schrödinger equations of the generalized planar pendulum were found in our work using supersymmetry methods where we also showed that the loci of the (conditionally quasi-exact) solvability coincide with the avoided (!) intersections of higher-lying potential energy surfaces. Further solutions were obtained in our work via the quantum Hamilton-Jacobi theory.
For the case of the topologic index k=2 and azimuthal quantum number m=1, the lowest two eigenenergies are analytic, see following table. All other states can be obtained only numerically.
Analytic energies | evaluated for β=10 |
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2 − β² | −98 |
4β − β² | −60 |
Note that η = 2β(m+1) and ζ = β² here.
Using FFT-DVR in angle Θ
Using Gauß-Legendre-DVR in cos Θ
Matlab version | C++ version |
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Animated sequence of eigenfunctions | Animated sequence of eigenfunctions |
Input data file | Input file and equivalent Python script |
Logfile output | Logfile output |