The three-dimensional stationary Schrödinger equation with potential

and with the reduced mass

can be written as

,

with the Laplacian

in spherical polar coordinates and the partial derivative

with respect to time.

With the three-dimensional isotonic or pseudoharmonic potential [3–5]:

.

The eigenfunctions can be expressed in spherical coordinates in a separable form:

.

The unnormalized radial

wavefunction is expressed in terms of the associated Laguerre polynomials

and the angular part is expressed in terms of spherical harmonics

.

The explicit bound state energies are obtained from (for more details see [5]):

.

To use the time dependence of the total phase function, the superposed wavefunctions Ψ are taken as a sum of just two eigenfunctions

, where the unnormalized wavefunction

for a particle is written as:

,

with

and integers

,

and

and, here, with

and

,

,

and with the remaining term in atomic units. In spherical polar coordinates, the total wavefunction becomes:

After a variable transformation from spherical polar coordinates into Cartesian coordinates, one obtain the phase function

. From the gradient of the phase

from the total wavefunction

in the eikonal (or polar) form

, with the quantum amplitude

, the velocity field

is calculated using:

.

For

or

, the velocity field

becomes autonomous and obeys the time-independent part of the continuity equation

with

; for

or

the trajectories reduce to circles with the velocities

and

, but with different signs:

,

where the velocity depends only on the position of the particle.

For the special initial positions (

,

and

) and with time

, the velocity vector becomes autonomous in the

and

directions, but not in the

direction:

;

For

, please refer to the code.

In this special case it is interesting to find initial positions where the

component of the velocity becomes negligibly small. This could be achieved for some values of the intersection of the equation

;

,

and

or for the trial solution with

,

,

.

In the very special cases the velocities vector in the

and

directions become autonomous (

) and

becomes negligibly small (

) for all times

. These orbits do not leave the

plane.

In all other cases, the velocity vector

becomes very complicated and the orbits appear to be unstable or ergodic. This means that as the trajectory evolves 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 (regular). 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 [6–8]. Further investigations are needed to get the full dynamics of the system.

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

.

The author would like to thank Ilka Weiß for helpful discussions.

[3] P. M. Davidson, "Eigenfunctions for Calculating Electronic Vibrational Intensities,"

* Proceedings of the Royal Society A*,

**135**(827), 1932 pp. 459–472.

doi:10.1098/rspa.1932.0045.

[5] 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.

[6] 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.

[7] A. C. Tzemos, G. Contopoulos and C. Efthymiopoulos,"Origin of Chaos in 3-D Bohmian Trajectories,"

*Physics Letters A, ***380**(45), 2016 pp.

* *3796–3802.

doi:10.1016/j.physleta.2016.09.016.