Bohm Trajectories for a Special Type of a Pseudoharmonic Potential

This Demonstration is the three-dimensional version of "Bohm Trajectories for an Isotropic Harmonic Oscillator with Added Inverse Quadratic Potential." It considers the Schrödinger equation that connects an isotropic harmonic oscillator potential with an additional centrifugal potential containing in the de Broglie–Bohm approach. This special pseudoharmonic-type potential has the form .
In de Broglie–Bohm theory, the particle has a well-defined trajectory in configuration space calculated from the total phase function. An appealing feature of the Bohmian description is that one can impose initial positions for the trajectories, just as in any classical dynamical system, in which the final positions of the particles are determined by their initial positions. Such initial positions are not controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. The wavefunction guides the particles in such a way that the particles avoid the regions in which the wave density is small and are attracted to the regions in which the wave density is large [1, 2]. In the regions with small wave density, the particles will be accelerated. The actual point-particles moving under the influence of the wavefunction are clearly seen by the oscillations of the two orbits in the graphics, which depends, among other things, on the parameter .
Exact solutions of the Schrödinger equation for this potential have the form of confluent hypergeometric functions. An analogous potential in three dimensions can represent the interaction of some diatomic molecules [3]. Obviously, the pseudoharmonic oscillator with 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.
The graphics show the wave density (if enabled), the velocity vector field (red arrows), the initial starting points of two possible orbits (red points, shown as small red spheres), the actual position (colored points, shown as small spheres) and two possible trajectories with the initial distance .

SNAPSHOTS

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DETAILS

The three-dimensional stationary Schrödinger equation with potential , a function only of the distance from the origin, can be written:
,
with the special pseudoharmonic-type potential
,
with the Laplace operator in spherical polar coordinates, the partial derivative with respect to time and with reduced mass , Planck's constant and the constants , . For simplicity, set , and equal to 1. The radial equation for isotropic harmonic oscillator plus inverse quadratic potential is solved by transformation into the confluent hypergeometric function to obtain a hyperradial solution (for more details see [3, 4]):
with , the integers , , and with the eigenenergy:
.
Due to the computational limitations, the superposed total wavefunctions consist here of just two eigenfunctions , where the unnormalized wavefunction for a particle, from which the trajectories are calculated, can be defined by:
with .
In spherical coordinates, the total wavefunction becomes:
.
From the wavefunction the velocity is calculated by the wave density flux :
,
where is the nabla operator and is the complex conjugate.
After a variable transformation from spherical polar coordinates into Cartesian coordinates, we obtain the velocity field .
For or , 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
,
where the velocity depends on the position of the particle only. For the velocities become . In all other cases the velocity vector becomes very complex and the orbits seem to be ergodic, which means here, when the trajectory proceeds in time the entire possible configuration space will be occupied by the orbit; that is, the orbits are dense everywhere. For some initial positions, the orbit will be closed and periodic. It is not obvious for this system with two superposed eigenfunctions that chaotic motion occurs, which would be associated with exponential divergence of initially neighboring trajectories [5].
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 .
References
[1] Bohmian-Mechanics.net. (Apr 24, 2020) bohmian-mechanics.net.
[2] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Apr 24, 2020) plato.stanford.edu/entries/qm-bohm.
[3] 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, 2012 pp. 1039–1059. doi:10.1007/s10910-011-9967-4.
[4] K. J. Oyewumi and E. A. Bangudu, "Isotropic Harmonic Oscillator Plus Inverse Quadratic Potential in N-Dimensional Spaces," The Arabian Journal for Science and Engineering, 28(2), 2003 pp. 173–182.
[5] 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.
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