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.