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

:

,

,

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

.

[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.