9464

WKB Computations on Morse Potential

The semiclassical Wentzel–Kramers–Brillouin (WKB) method applied to one-dimensional problems with bound states often reduces to the Sommerfeld–Wilson quantization conditions, the cyclic phase-space integrals . It turns out that this formula gives the exact bound-state energies for the Morse oscillator with . The requisite integral can be reduced to , in which and are the classical turning points . The integral can be done "by hand", using the transformation followed by a contour integration in the complex plane, but Mathematica can evaluate the integral explicitly, needing only the additional fact that The result reads
, which can be solved for to give , , in units with . The highest bound state is given by , where represents the floor, which for positive numbers is simply the integer part. The values of , , and (expressed in atomic units) used in this Demonstration are for illustrative purposes only and are not necessarily representative of any actual diatomic molecule.

SNAPSHOTS

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

DETAILS

A particle in a one-dimensional potential can be described by the Schrödinger equation: .
The semiclassical or WKB method is based on the ansatz . In the limit as , in the exponential satisfies the Hamilton–Jacobi equation for the action function , with one solution . It is then shown in most graduate-level texts on quantum mechanics (e.g. Schiff, Merzbacher, etc.) that this usually leads to the Sommerfeld–Wilson quantum conditions on periodic orbits , . For one-dimensional problems, the cyclic integral can be replaced by , where , are the classical turning points of the motion and .
As an étude consider the linear harmonic oscillator, with and . The quantum condition reads , which can be solved to give . This is one of a small number of cases in which the WKB method gives the exact quantum-mechanical energies.
    • 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+