The Three-Dimensional Isotonic Potential in Bohmian Mechanics

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In quantum chemistry the study of exact solutions of the Schrödinger equation for an harmonic oscillator with a centripetal barrier, called the three-dimensional isotonic oscillator or pseudoharmonic potential, is of considerable interest. Diatomic molecules contain two atoms that are chemically bonded; homonuclear if the two atoms are identical, such as the oxygen molecule , heteronuclear if the atoms are different, as carbon monoxide . The three-dimensional version in spherical polar coordinates of the isotonic potential for a heteronuclear diatomic molecule in the de Broglie–Bohm approach (often called Bohmian mechanics or causal interpretation of quantum mechanics) is considered here.


In the 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 on can impose initial positions on the trajectories of the particles, just as in any classical dynamical system. The final positions are then determined by their initial positions. These 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 [1, 2]. In regions with small wave density, the particles will be accelerated.

This Demonstration investigates the de Broglie–Bohm approach for an isotonic potential in a superposition eigenstate .

The potential is written (in atomic units) as:


is the dissociation energy between two atoms and is the equilibrium intermolecular separation constant.

The graphic shows the wave density (if enabled), the velocity vector field (red arrows, if enabled in the code), the initial starting points of eight possible orbits (yellow points, shown as small yellow spheres), the actual position (colored points, shown as small spheres) and two possible trajectories with the initial distance .


Contributed by: Klaus von Bloh (May 2021)
Open content licensed under CC BY-NC-SA



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 radial term is:



as the radial part.

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


for example,


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.


[1] (Apr 8, 2021)

[2] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Apr 8, 2021)

[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.

[4] Y. Weissman and J. Jortner, "The Isotonic Oscillator," Physics Letters A, 70(3), 1979 pp. 177–179. doi:10.1016/0375-9601(79)90197-X.

[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.

[8] K. von Bloh. The Three-Dimensional Isotonic Potential in the Bohmian Mechanics Approach. [Video]. (Apr 8, 2021)

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