Degenerate Eigenstates

In quantum mechanics, if any eigenstate is -fold degenerate, there are an infinite number of choices for the orthogonal eigenfunctions. The simplest possible example is the free particle in one dimension. Every energy level is twofold degenerate. This corresponds to the physical fact that particles moving in opposite directions have the same kinetic energy. The Schrödinger equation has two linearly independent eigenfunctions. A common choice takes . These functions are delta function-normalized, such that , and are also eigenfunctions of linear momentum , with the eigenvalues .
In this Demonstration, you can explore any of an infinite number of orthonormalized degenerate pairs of eigenfunctions, which can be represented by and , where . Both real and imaginary parts of each wavefunction are plotted.



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


Snapshots 1 and 2: varying values of
Snapshot 3: for , the eigenfunctions are proportional to and
Reference: S. M. Blinder, Introduction to Quantum Mechanics, Amsterdam: Elsevier, 2004 pp. 31–32.
    • 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.

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 © 2018 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+