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Associated Legendre polynomials arise as the solution of the Schrödinger equation

, with , , , , and so on. A degenerate, unnormalized wavefunction for the two-dimensional perturbed case, which leads to a transition from the state to the state , can be expressed by

,

where , are eigenfunctions, and are permuted eigenenergies of the corresponding stationary one-dimensional Schrödinger equation, with and . In the wavefunction , the perturbation term is given by , where the parameter is an arbitrary constant. The eigenfunctions , are defined by

,

where , are associated Legendre polynomials. are the quantum numbers with and . The wavefunction is taken from [3].

For , the velocity field obeys the time-independent part of the continuity equation with , where is the complex conjugate. For this special case (), the trajectory becomes strongly periodic, because of the sign changing for with of the velocity term.

In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps are increased (if enabled), the results will be more accurate.

References

[1] C. Dewdney and M. M. Lam, "What Happens during a Quantum Transition?," Information Dynamics (H. Atmanspacher and H. Scheingraber, eds.), New York: Plenum Press, 1991.

[2] C. Efthymiopoulos, C. Kalapotharakos, and G. Contopoulos, "Origin of Chaos near Critical Points of the Quantum Flow," Physical Review E, 79, 2009 036203. doi:10.1103/PhysRevE.79.036203, arXiv:0903.2655 [quant-ph].

[3] M. Trott, The Mathematica GuideBook for Symbolics, New York: Springer, 2006.

[4] G. Pöschl and E. Teller, "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators," Zeitschrift für Physik, 83 (3–4), 1933 pp. 143–151, doi:10.1007/BF01331132.

[5] "Bohmian-Mechanics.net." (Jul 30, 2015) www.bohmian-mechanics.net/index.html.

[6] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jul 30, 2015)plato.stanford.edu/entries/qm-bohm.