9716

Simple Chaotic Motion of Quantum Particles According to the Causal Interpretation of Quantum Theory

The causal interpretation of quantum theory developed by David Bohm introduced trajectories that are guided by a real phase function from the wavefunction in the polar form. A simple model is used to get chaotic motion; the trajectories undergo a transition from order to chaos depending on the relative phase factors and . The trajectories circle close to the minimum (nodal) points. For example, at the origin the squared wavefunction has a minimum for all : . The graphic shows the squared time-dependent wavefunction, the quantum particles, and the complete paths.

SNAPSHOTS

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

DETAILS

Chaotic motion takes place when a relative phase is introduced for a superposition of two stationary eigenstates of an isotropic harmonic quantum oscillator.
The Schrödinger equation, with , is .
The unnormalized wavefunction used is
, with the eigenfunctions and the energy , with .
The polar form of the wavefunction is.
The velocities and therefore the trajectories are derived from the gradient of the phase function .
The model is based on a paper by A. Makowski and M. Frackowiak (arXiv:quant-ph/0111155v1 (2001)).
    • 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 © 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+