9827

Bohm Trajectories for Quantum Particles in a Uniform Gravitational Field

This Demonstration studies a free-falling quantum particle with mass in a uniform gravitational potential (in the Earth's gravitational field: ). The equations of motion can be solved analytically for the classical and quantum-mechanical case. For the classical (i.e., Newtonian) case, the force is , which leads to the kinematic equation of motion: with constant acceleration , constant initial velocity , and initial position . Classically, the kinematic equation is independent of the mass, which shows that all bodies with different masses fall with an equal acceleration . In the quantum case the time-evolution of the wavefunction for an initial Gaussian packet (initial maximum , group velocity , initial density , mass ) in a uniform field with the strength is described by a complex-valued field:
, satisfying Schrödinger’s wave equation (here: ). The center of the packet accelerates in the direction . Bohm's causal interpretation (today often called the de Broglie-Bohm interpretation) of non-relativistic quantum mechanics leads to exactly the same experimental predictions as conventional interpretations and it is a fully deterministic theory of particles in motion (not directly measurable), but a motion of a profoundly non-Newtonian sort. From the phase of the wavefunction in the eikonal form , the velocity in the configuration space is found by , with
.
Integrating with respect to time the analytic result for is , where is the integration constant, which is used to estimated the positions of the initial particles. The quantum particle acceleration is , which is compared with the classical particle acceleration . Considering both cases, two differences are obvious: First, the quantum motion is a superposition of classical terms (with , the group velocity) and a term due to the spreading of the packet, which is not affected by the external field. Second, the quantum motion in the uniform field does depend on the mass. When the particle lies initially at the center of the squared wavefunction () the motion becomes classical. In the causal interpretation, the effective potential is the sum of quantum potential and potential . On the right side, the graphic shows the squared wavefunction and the trajectories. On the left side, you can see the position of the particles, the squared wavefunction (blue), the quantum potential (red), and the velocity (green). The velocity is scaled down.

SNAPSHOTS

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

DETAILS

Reference:
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+