# Bohm Trajectories for an Isotropic Harmonic Oscillator with Added Inverse Quadratic Potential

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This Demonstration considers two-dimensional Bohm trajectories in a central field represented by an isotropic harmonic oscillator augmented by an inverse quadratic potential plus a constant . This is called a pseudoharmonic-type potential, with the form . Exact solutions of the Schrödinger equation for this potential are known. An analogous potential in three dimensions can represent the interaction of some diatomic molecules [1]. Obviously, the pseudoharmonic oscillator behaves asymptotically as a harmonic oscillator, but has a singularity at For , there is a small region where the potential exhibits a repulsive inverse-square-type character. Possible trajectories can then exhibit a rich dynamical structure, depending on the parameters , of the potential and the initial starting points. The motion ranges from periodic to quasi-periodic to fully chaotic. In the de Broglie–Bohm (or causal) interpretation of quantum mechanics [2, 3], the particle position and momentum are well defined, and the motion can be described by continuous evolution according to the time-dependent Schrödinger equation. In contrast to [4], the conditions for chaotic behavior can occur in a system with two degrees of freedom and for a superposition of two stationary states. See, for example, , (in the variable in the program), , , , and the initial position , two trajectories with initial distance .

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Contributed by: Klaus von Bloh (October 2018)

Open content licensed under CC BY-NC-SA

## 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.

## Snapshots

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