9853

Time-Dependent Superposition of Harmonic Oscillator Eigenstates

Consider a time-dependent superposition of quantum harmonic oscillator eigenstates, , where the eigenfunctions and eigenvalues are given by and , respectively. Here is the Hermite polynomial. The Hamiltonian for this system is and its energy expectation value is given by .
Choosing "energy levels" shows the complex wavefunction in the upper panel, where the shape is its modulus and the coloring represents its argument, the range to corresponding to colors from red to magenta. The potential energy curve is drawn for visualization purposes. The lower panel shows the eigenvalues in blue and the energy of the superposition state in red.
Choosing the view "expectation value" shows the same superposition in both the position and momentum representations, where the wavefunctions are connected via . The two left panels show the position space probability density and position expectation value , while the right panels show the momentum space probability density and momentum expectation value . The lower panel shows a parametric plot of the expectation values which, in the very special case of a harmonic potential, resembles classical phase space.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
    • 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+