# Bohm Trajectories for the Two-Dimensional Coulomb Potential

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

For quantum mechanics in two dimensions, the most advantageous choice of coordinates is determined by the form of the potential. This Demonstration considers the two-dimensional reduction of the three-dimensional Schrödinger equation of the hydrogen atom in the de Broglie–Bohm interpretation of quantum mechanics, using polar coordinates.

[more]
Contributed by:Klaus von Bloh (May 2018)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

In plane polar coordinates (, ), the Schrödinger equation for the two-dimensional hydrogen-like atom reads

,

with the Coulomb potential

,

the reduced mass , a constant and with , and so on.

With the ansatz

,

the Schrödinger equation then separates to become

,

.

is the radial part of the wavefunction with the eigenenergy term , and is the single-valued eigenfunction of the angular momentum operator; the integer is set by (), and is the eigenenergy. The solutions of the Schrödinger equation for the radial part are the associated Laguerre polynomials with a retracted definition of the radius :

with . For this special case, , and are given by:

,

,

,

with and .

The unnormalized stationary wavefunction could be written in the form:

.

For this Demonstration, a superposition of two eigenstates with a constant phase shift factor is studied:

,

which leads in Cartesian coordinates (but abbreviating , ) to the wavefunction with :

.

In the wavefunction , the perturbation term is given by , where the parameters and are arbitrary constants.

For , the squared wavefunction (particle density ) becomes in Cartesian coordinates, where is the complex conjugates:

From the wavefunction for , the equation for the phase function follows:

and therefore for the components of the velocity:

For , the velocity field becomes autonomous and obeys the time-independent part of the continuity equation

with . For the special case , the trajectories reduce to circles with the velocities and :

and .

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] Bohmian-Mechanics.net. (Mar 2, 2018) www.bohmian-mechanics.net/index.html.

[2] S. Goldstein. "Bohmian Mechanics." *The Stanford Encyclopedia of Philosophy (Summer 2017 Edition)*. (Jan 30, 2018)plato.stanford.edu/entries/qm-bohm.

[3] A. Cesa, J. Martin and W. Struyve, "Chaotic Bohmian Trajectories for Stationary States," *Journal of Physics A: Mathematical and Theoretical*, 49(39), 2016 395301. doi:10.1088/1751-8113/49/39/395301. arXiv:01387v2 [quant-ph].

[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(96)00096-5.

[5] C. Efthymiopoulos, C. Kalapotharakos and G. Contopoulos, "Nodal Points and the Transition from Ordered to Chaotic Bohmian Trajectories," *Journal of Physics A: Mathematical and Theoretical*, 40(43), 2007 12945. doi:10.1088/1751-8113/40/43/008.

## Permanent Citation