Trajectories for a Particle in a TwoDimensional Circular Billiard This Demonstration considers two trajectories of a quantum particle in a twodimensional 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
.
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Figure 1: Two Trajectories in (x-y)- configuration space
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, indices
,
, where and
,where
is the
zero of
and the
are integers. The energies are given by:
is the mass of the particle.
The eigenfunction in polar coordinates:
obeys the corresponding stationary Schrödinger equation ,
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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. RELATED LINKS Bohm, David Joseph (1917–1992) (ScienceWorld) Schrödinger Equation (Wolfram MathWorld) Spherical Bessel Function of the First Kind (Wolfram MathWorld) Separation of Variables (Wolfram MathWorld) Bessel Function Zeros (Wolfram MathWorld) Polar Coordinates (Wolfram MathWorld) Quasiperiodic Motion (Wolfram MathWorld) Particle in an Infinite Circular Well (Wolfram Demonstrations Project) From Bohm to Classical Trajectories in a Hydrogen Atom (Wolfram Demonstrations Project) Chaos (Wolfram MathWorld) 3
Nodal Points in Bohmian Mechanics (Wolfram Demonstrations Project) Influence of a Moving Nodal Point on the "Causal Trajectories" in a Quantum Harmonic Oscillator Potential (Wolfram Demonstrations Project) Influence of the Relative Phase in the de Broglie-Bohm Theory (Wolfram Demonstrations Project) The Causal Interpretation of the Triangular Quantum Billiard (Wolfram Demonstrations Project) PERMANENT CITATION
Contributed by: Klaus von Bloh
After work by: O. F. de Alcantara Bonfim, J. Florencio and F. C. Sá Barreto
"Bohm Trajectories for a Particle in a Two-Dimensional Circular Billiard" http://demonstrations.wolfram.com/BohmTrajectoriesForAParticleInATwoDimension alCircularBilliar/ Wolfram Demonstrations Project Published: August 11, 2017
Contributed by: Klaus von Bloh
Betreff: Your submission to the Wolfram Demonstrations Project Absender: Wolfram Demonstrations Project Empfänger:
[email protected] Datum: 26. August 2017 13:30
Dear Klaus von Bloh, We are happy to inform you that your submission Bohm Trajectories for a Particle in a TwoDimensional Circular Billiard to the Wolfram Demonstrations Project has been accepted for publication. Your Demonstration will now be available to all visitors to the Wolfram Demonstrations Project site. The permanent URL for your Demonstration is: http://demonstrations.wolfram.com/BohmTrajectoriesForAParticleInATwoDimensionalCircular Billiar/ It will be available within the next 24 hours.
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