9887

A Breather Solution in the Causal Interpretation of Quantum Mechanics

A breather solution appears often in nonlinear wave mechanics, in which a nonlinear wave has energy concentrated in a localized oscillatory manner. This Demonstration studies a breather solution with a hyperbolic secant envelope of the focusing nonlinear Schrödinger (NLS) equation , with and so on, also known as the Gross–Pitaevskii equation in the causal interpretation, developed by Louis de Broglie and David Bohm.
The NLS equation could be interpreted as the Schrödinger equation (SE) with the nonlinear potential term with , although for most situations it has no relationship with the quantum Schrödinger equation other than in name. In the studied breather, there are large density amplitudes at certain times, which could be interpreted as rogue waves.
The graphic on the left shows the density (blue), the quantum potential (red), and the velocity (green). On the middle and on the right, you can see the density and the trajectories in space and the quantum potential and the trajectories in space. The velocity and the quantum potential on the left side are scaled to fit.

SNAPSHOTS

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

DETAILS

One of the exact breather solutions for is a complex field as a function of and which is periodic in space .
The complex-valued wavefunction
, with ,
has several remarkable features (see [1]), including periodicity of the squared wavefunction with period implicit analytic solutions of quantum trajectories for some values a time-independent solution for and , and a symmetry according to , where , with . For the case , the wavefunction becomes , where an implicit analytic solution for the quantum motion is given by the gradient of the phase function of the wavefunction in the eikonal form (often called polar form). For this special case, the implicit function for is , with the integration constant .
Exploiting to the symmetry of the wave density, only one half of the trajectories were calculated numerically.
On YouTube there are some videos by the author, which show additional breather solutions with a hyperbolic secant envelope in the de Broglie–Bohm interpretation for the Gross–Pitavevskii equation.
References
[1] D. Schrader, Asymptotisch Auftretende Solitonen-Lösungen der Nichtlinearen Schrödinger-Gleichung zu Beliebigen Secans-Hyperbolicus-Förmigen Anregungen, Aachen, Germany: Shaker Verlag, 1998.
[2] Bohmian-Mechanics.net. (Jul 11, 2014) www.bohmian-mechanics.net/index.html.
[3] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jul 11, 2014) 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 © 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+