9860

Closure Property of Eigenfunctions

A complete set of discrete eigenfunctions obeys the orthonormalization conditions . Complementary to these is the set of closure relations . For real eigenfunctions, the complex conjugate can be dropped. The finite sums for up to 100 are evaluated in this Demonstration. Four systems are considered: (1) infinite square-well potential with and eigenfunctions , , ; (2) linear harmonic oscillator with and , , ; (3) linear rigid rotor for fixed , , , , ; (4) hydrogen-like radial function with , , , .
For the first three cases, the sum approaches an oscillatory representation of the delta function . However, the hydrogenic functions represent only the discrete bound states. They do not constitute a complete set of eigenfunctions without including the continuum. The sums usually exhibit erratic behavior but sometimes do show a peaking, particularly for larger values of .

SNAPSHOTS

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

DETAILS

The closure relation can be derived by considering the expansion of an arbitrary function obeying the same analytic and boundary conditions as the eigenfunctions . If the set of eigenfunctions is complete, one can write , with expansion coefficients determined from . Substituting the last relation into the expansion, we find , with the summation equivalent to the delta function .
Reference: Any graduate-level text on quantum mechanics.
    • 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+