# Bohm Trajectories for the Two-Dimensional Coulomb Potential

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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.

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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.