Bohm Trajectories for a Particle in a Two-Dimensional Circular Billiard
This Demonstration considers two trajectories of a quantum particle in a two-dimensional configuration space in which the particle is trapped in a "circular billiard potential" . The trajectories of the particle can then exhibit a rich dynamical structure. The motion ranges from periodic to quasi-periodic to fully chaotic. In the de Broglie–Bohm (or causal) interpretation of quantum mechanics [2, 3], the particle position and momentum are well defined, and the motion can be described by continuous evolution according to the time-dependent Schrödinger equation. The nodal point near the circular origin and the wave density restrict the motion of the particle to a region around the center of the billiard.
In Bohm’s approach to quantum mechanics, the signature of chaos can be obtained in the same way as in classical mechanics: chaotic motion means the exponential divergence of initially neighboring trajectories.
Chaos emerges from the nature of the quantum trajectory around the nodal point, which, in turn, depends on the locations and the frequencies of the quantum particle.
Due to the periodicity of the superposed wavefunction in space, the initial position and the influence of the constant phase factor, the trajectories could leave the billiard regime for certain time intervals. To ensure that the trajectories lie within the billiard regime, the initial points are restricted to an area: and .
The graphic shows the trajectories (blue/black), the velocity vector field (red), the nodal point (red), the absolute value of the wavefunction, the boundary of the circular box (dashed white line), and the initial (shown as a big blue point with a white square you can drag or black points) and final points of the trajectories (shown as small blue/black points). The initial distance between the two starting points is determined by the factor .
Consider a particle in a two-dimensional circular box of radius . The eigenfunctions of the system are expressed in terms of Bessel functions of the first kind,
where is the zero of and the indices and are integers. The energies are given by:
where is the mass of the particle.
The eigenfunction obeys the corresponding stationary Schrödinger equation in polar coordinates:
where is the central potential energy with
For simplicity, set the mass and .
An unnormalized wavefunction for a particle, from which the trajectories are calculated, can be defined by a superposition state:
with , which is a slightly different approach than .
For the starting spatial point , the velocity field has a singularity.
The results will be more accurate by increasing PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps in the code.
 O. F. de Alcantara Bonfim, J. Florencio and F.C. Sá Barreto, "Chaotic Bohm’s Trajectories in a Quantum Circular Billiard," Physics Letters A,277(3), 2000 pp. 129–134. doi:10.1016/S0375-9601(00)00705-2.