Details

The two-dimensional stationary Schrödinger equation with potential , a function only of the distance from the origin, can be written:

with the potential

,

reduced mass , Planck's constant , the constants and the Laplacian operator in plane polar coordinates. For simplicity, set and equal to 1. The solution of the Schrödinger equation for the quantum system with the pseudoharmonic-type potential gives to the eigenfunction (for detailed information, see [5]) in plane polar coordinates:

with , the integer and with the eigenenergy :

In Cartesian coordinates (, ), the solution reads:

.

or in Cartesian coordinates:

For , the velocity field becomes autonomous and obeys the time-independent part of the continuity equation

with ; the trajectories reduce to circles with the velocities and :

and .

When PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps are increased (if enabled), the results will be more accurate. The initial distance between the two starting trajectories is determined by the factor . In the program, you can change the parameters of the potential.

References

[1] K. J. Oyewumi and K. D. Sen,"Exact Solutions of the Schrödinger Equation for the Pseudoharmonic Potential: An Application to Some Diatomic Molecules," Journal of Mathematical Chemistry, 50(5), 2012 pp. 1039–1059. doi:10.1007/s10910-011-9967-4.

[2] Bohmian-Mechanics.net. (Oct 8, 2018) www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/index.html.

[3] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Oct 2, 2018)plato.stanford.edu/entries/qm-bohm.

[4] R. H. Parmenter and R. W. Valentine, "Deterministic Chaos and the Causal Interpretation of Quantum Mechanics," Physics Letters A, 201(1), 1995 pp. 1–8. doi:10.1016/0375-9601(95)00190-E.

[5] S. M. Ikhdair and R. Sever, "Exact Solutions of the Radial Schrödinger Equation for Some Physical Potentials," Central European Journal of Physics, 5(4), 2007 pp. 516–527. doi:10.2478/s11534-007-0022-9.