# Bohm Trajectories for the Two-Dimensional Coulomb Potential

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.

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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 .

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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.

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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.