11405

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" [1]. 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 .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

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 [1].
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.
References
[1] 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.
[2] "Bohmian-Mechanics.net." (Jul 28, 2017) www.bohmian-mechanics.net/index.html.
[3] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jul 28, 2017) plato.stanford.edu/entries/qm-bohm.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+