Details

The three-dimensional stationary Schrödinger equation with potential can be written as

,

with the Hartmann potential

,

here with and ;

with the Laplace operator in spherical polar coordinates and the partial derivative with respect to time. The unnormalized radial wavefunction is expressed in terms of the associated Laguerre polynomials and the angular part expressed in terms of the Jacobi polynomials . The energy spectrum equation depends only on the integers and and the parameters and :

with

as the radial part.

with

as the angular part, with and integers and and with the eigenenergies given by:

.

The complete eigenfunction is thus (for more details see [3]):

.

For the noncentral Hartmann potential, no degenerate quantum states are possible. Due to computational limitations, the superposed total wavefunctions are taken as a sum of just two eigenfunctions , where the unnormalized wavefunction for a particle is written as:

,

with , and .

In spherical polar coordinates, the total wavefunction becomes:

After a variable transformation from spherical polar coordinates into Cartesian coordinates, we 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:

.

If the coupling term becomes large, the quantum orbits are carried away from the center. The closer the starting points are to the axis, the faster the particles rotate.

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 :

,

and

,

where the velocity depends on the position of the particle only. For or , the velocities become . In all other cases, the velocity vector becomes very complex and the orbits appear to be unstable or ergodic. This means here that when 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 [4, 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] H. Hartmann, "Die Bewegung eines Körpers in einem ringförmigen Potentialfeld," Theoretica chimica acta, 24(2), 1972 pp. 201–206.doi:10.1007/BF00641399.

[2] A. Arda and R. Sever, "Non-Central Potentials, Exact Solutions and Laplace Transform Approach," Journal of Mathematical Chemistry, 50(6), 2012 pp. 1484–1494. doi:10.1007/s10910-012-9984-y.

[3] Ö. Ye\:015filta\:015f and R. Sever, "Exact Solutions of Schrodinger Equation for a Ring-Shaped Potential." arxiv.org/abs/quant-ph/0703034.

[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] 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.

[6] Bohmian-Mechanics.net. (Aug 19, 2020) bohmian-mechanics.net.

[7] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Aug 19, 2020) plato.stanford.edu/entries/qm-bohm.