Bohm Trajectories for the Noncentral Hartmann Potential
A noncentral potential in spherical polar coordinates depends on the angular variables and [1, 2]. The Hartmann potential , depending on the variables and , was introduced in quantum chemistry  to describe ring\[Hyphen]shaped molecules, such as benzene. It can also be applied to the interaction between distorted nuclei. The Hartmann potential adds to the three-dimensional Kepler–Coulomb potential term, which corresponds to coupling with the radial degree of freedom  with a coupling constant .[more]
The minimum of the potential well , the coupling constant and the radial distance of the potential minimum from the center of the potential ring are three system-specific variables. The motion of a quantum particle under the influence of the Hartmann potential can be solved exactly in closed form, in terms of Laguerre and Jacobi polynomials for radial and angular parts, respectively.
In this Demonstration, the de Broglie–Bohm approach for a Hartmann potential in a superposition eigenstate is investigated. The potential is written in the following form, in atomic units:
For , the potential reduces to the three-dimensional Kepler–Coulomb potential of the hydrogen atom.
In de Broglie–Bohm theory, the particle has a well-defined trajectory in configuration space, calculated from the gradient of the phase function. An appealing feature of the Bohmian description is that one can impose initial positions for the trajectories, just as for a 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 are attracted to the regions in which the wave density is large [6, 7]. In the regions with small wave density, the particles will be accelerated.
The graphics show the wave density (if enabled), the velocity vector field (red arrows), the initial starting points of eight possible orbits (red points, shown as small red spheres), the actual position (colored points, shown as small spheres) and eight possible trajectories with the initial distance .[less]
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 :
as the radial part.
as the angular part, with and integers and and with the eigenenergies given by:
The complete eigenfunction is thus (for more details see ):
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 :
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 .
 H. Hartmann, "Die Bewegung eines Körpers in einem ringförmigen Potentialfeld," Theoretica chimica acta, 24(2), 1972 pp. 201–206.doi:10.1007/BF00641399.
 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.
 Ö. Ye\:015filta\:015f and R. Sever, "Exact Solutions of Schrodinger Equation for a Ring-Shaped Potential." arxiv.org/abs/quant-ph/0703034.
 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.
 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.
 Bohmian-Mechanics.net. (Aug 19, 2020) bohmian-mechanics.net.
 S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Aug 19, 2020) plato.stanford.edu/entries/qm-bohm.