9722

Wave Packet Dynamics

Wave packet dynamics can be studied by pump-probe femtosecond spectroscopy of vibrations of molecules in excited states (see, e.g. A. Assion et al., Z. Phys. D, 36, 1996 pp. 265–271). As a simple example, consider a superposition of the lowest three eigenstates of the harmonic oscillator. The initial state is set by choosing the relative amplitudes a, b, and c of levels 0, 1, and 2. The Demonstration shows the system's time development in a static harmonic oscillator potential.

THINGS TO TRY

SNAPSHOTS

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

DETAILS

Mathematically, a wave packet is an integrable superposition of the wave functions of the stationary problem with time-dependent development coefficients, , where is an eigenfunction of the stationary Schrödinger equation. The time-development coefficient in the case of a stationary potential is given by .
Based on the harmonic oscillator and a superposition of the lowest three eigenstates, this can be written as , with .
A simplified wave function is given using the appropriate Hermite polynomial :
.
The recurrence times are determined by the lowest level of the harmonic oscillator that changes most slowly. Because the harmonic oscillator levels are equidistant, the example is especially simple, showing a strict periodic behavior. For the Demonstration real relative amplitudes , , and were chosen for didactic reasons, although in general the relative amplitudes may be complex. In the figure, the real part, the imaginary part, or the probability density of the lowest three harmonic oscillator levels ( = 0, 1, 2) is shown, together with the resulting total wave function.
    • 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+