9873

Bohm Trajectories for Quantum Particles in a Time-Dependent Linear Potential

The time-dependent probability density associated with this Gaussian wave packet is Gaussian for all times. The de Broglie–Bohm formulation of quantum mechanics, also known as the quantum theory of motion, is a single-valued theory in configuration space with one possible velocity ( is momentum) for a given position at time . A point particle follows a trajectory given by the equation of motion. In this Demonstration the time-dependent potential is assumed to be . Therefore, describes a normalized Gaussian wave packet that is initially centered at . The shape of the wave packet is not changed by the external force. The particle trajectories are obtained via the velocity field (adopting ), where is the phase of the wave function in the eikonal form . Their time evolution can be given in closed form: , where , , and are arbitrary real constants, and where , the integration constant, is used to estimated the positions of the initial particles. For the classical case the equation of motion can be solved analytically from Newton's second law : , which is considered with the quantum motion. The quantum particle dynamics is a sum of classical terms (with , but without the -term) and a term due to the spreading of the packet. When the particle lies initially at the center of the squared wavefunction () the motion becomes classical. You can calculate the wavefunction, the gradient of the phase (), and the analytic solution of the quantum motion for any arbitrary time-dependent function , if the terms are integrable. The graphics show the squared wavefunction and the trajectories on the right, and the position of the particles, the squared wavefunction (blue), and the quantum potential (red) on the left.

SNAPSHOTS

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

DETAILS

The time evolution of a quantum particle in the presence of a time-dependent linear potential is given by a complex-valued function
that satisfies Schrödinger's wave equation .
References:
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.
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+