9464

Nodal Points in Bohmian Mechanics

Moving vortices or nodal points play an important role in testing the attributes of quantum motion in the framework of the de Broglie–Bohm trajectory method [2]. This Demonstration studies an unnormalized superposed wavefunction for the two-dimensional harmonic oscillator. Chaos emerges from the sequential interaction between the quantum path with the moving nodal points, depending on the distance and the frequencies between the quantum particles and their initial positions. Here, chaotic motion means the exponential divergence of initially neighboring trajectories (see Related Links below). Vortices are readily formed in several different scenarios, including the superposition of quantum states [1, 2]. Although the initial quantum state contains only one or three nodal points, depending on the value, during the time evolution new vortices are created and annihilated, so that at certain times there are at most five vortices present. If , there is a nodal region and the squared wavefunction becomes time-independent. In the vicinity of the vortices, the velocity vector field becomes circular and the trajectory is very unstable. The trajectories of the vortices are periodic. The Bohmian trajectory forms periodic, quasi-periodic, or chaotic curves while interacting with the nodal points. The graphic shows the trajectory (white), the velocity vector field (red), the nodal points (blue), the absolute wavefunction, and the initial and end point (white) of the trajectory. You can return to the original settings with the "initialize" checkbox.

SNAPSHOTS

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

DETAILS

The unnormalized wavefunction
satisfies the Schrödinger equation and is taken from [1]:
,
with , , and so on. The velocity is deduced from the total phase of the wavefunction in the eikonal form [3]. For , there exists an analytic solution. The corresponding autonomous differential equation system (velocity field) derived from the phase of the total wavefunction is
(in the direction) and (in the direction), which can be integrated analytically with respect to time to yield the motion in the - configuration space, which leads, with integration constants and to
and
.
References
[1] A. Klein, D. Jaksch, Y. Zhang, and W. Bao, "Dynamics of Vortices in Weakly Interacting Bose-Einstein Condensates," Physical Review A, 76(4), 2009 pp. 043602–1-043602-7. doi:10.1103/PHYSREVA.76.043602; arXiv: 0709.2132v1 [quant-ph].
[2] C. Efthymiopoulos, C. Kalapotharakos, and G. Contopoulos, "Origin of Chaos near Critical Points of the Quantum Flow," Physical Review E, 79, 2009 pp. 036203–1–036203–18. doi:10.1103/PhysRevE.79.036203; arXiv:0903.2655 [quant-ph].
[3] P. Holland, The Quantum Theory of Motion, Cambridge: Cambridge University Press, 1993.
    • 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.
Powered by Wolfram Mathematica © 2014 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+