Bohm Trajectories for the Two-Dimensional Coulomb Potential

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.
In the de Broglie–Bohm interpretation, the particle position and momentum are well defined, and the motion can be described as a continuous evolution of the quantum particle according to the time-dependent Schrödinger equation [1, 2]. The two-dimensional system is the Bohmian mechanics analog of the Bohr–Sommerfeld model. The Bohr–Sommerfeld model produces circular or elliptical orbits, analogous to those in Kepler's laws of planetary motion. In contrast, the Bohm model allows a wide variety of orbits. The trajectories vary between periodic and chaotic motion, as determined by the factor . A dynamical system is chaotic if it displays strong sensitivity to initial conditions.
The nodes of the wavefunction, that is, points or domains where the wavefunction equals zero, play an important role in the generation of chaos, but little is known about the possibility of chaos in the case of stationary nodal points [3] or domains. Moving nodal points and at least three stationary states with at least one pair of these states having mutually incommensurate energy eigenvalues are, in general, responsible for chaotic orbits of quantum particles [3–5].
In this Demonstration, the total wavefunction is in a very special superposition of two stationary states with a constant phase shift , in which the second term is the perturbation term controlled by the factor .
In this case, the nodal point is stationary and the nodal domain is determined by the factor and the perturbation factor . In nodal domains, the particles are highly accelerated, which leads to an exponential separation of initially neighboring trajectories.
This Demonstration shows that the nodes do not need to move in order to generate chaos. In contrast to [4], the conditions for chaotic behavior can occur in a system with two degrees of freedom and for a superposition of two stationary states.
The graphic shows the trajectories (white/blue), the velocity vector field (red), the nodal point (blue), the absolute value of the wavefunction, and the initial (shown as a big white point with a white square you can drag or black points) and final points of the trajectories (shown as small white/blue points). The quantum potential (if enabled) is shown as blue/black contour lines.

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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.
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