Demos.GeneralPendulum.Spherical

Generalized pendulum (spherical)

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.

Solutions for k=2 and m=1

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
2 − β²	−98
4β − β²	−60

Note that η = 2β(m+1) and ζ = β² here.

Using FFT-DVR in angle Θ

• Animated sequence of eigenfunctions
• Input data file
• Logfile output

Using Gauß-Legendre-DVR in cos Θ

Matlab version	C++ version
Animated sequence of eigenfunctions	Animated sequence of eigenfunctions
Input data file	Input file and equivalent Python script
Logfile output	Logfile output