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

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 .
The graphic shows the trajectories (white/blue), the velocity vector field (red), the absolute value of the wavefunction and the initial and final points of the trajectories. The initial point is shown as a white square, which you can drag. The final point is shown as a small white or blue dot. The pseudoharmonic-type potential (if enabled) is shown with blue/black contour lines.

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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:
.
An unnormalized wavefunction for a particle, from which the trajectories are calculated, can be defined by a superposition state :
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.
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