9459

Supersymmetry for the Square-Well Potential

The most elementary problem in quantum mechanics considers a particle of mass in a one-dimensional infinite square well of width ("particle in a box"). The Schrödinger equation can conveniently be written in the modified form in , such that the ground state energy is rescaled to . The eigenstates are then given by -1], . The quantum number is now equal to the number of nodes in the wavefunction. For simplicity, let and . The Schrödinger equation then simplifies to with , , , .
The first step is to define the superpotential and two ladder operators and . The original Hamiltonian is then given by . The operator obtained by reversing and , , is called the supersymmetric-partner Hamiltonian. More explicitly, and , where and . It can then be shown that if is an eigenfunction of with eigenvalue then is an eigenfunction of with the same eigenvalue: . We denote the eigenfunction of by call its eigenvalue . For unbroken supersymmetry, . Note that , meaning that the ground state of has no superpartner. Correspondingly, we find . (The constants provide normalization factors.) Note that the operator removes one of the nodes of the wavefunction as it converts it into . Conversely, adds a node.
In this Demonstration, you can plot any of the lowest four square-well eigenfunctions , on a scale with each origin at the corresponding eigenvalue . On the right are the corresponding eigenfunctions of the supersymmetric partner Hamiltonian , moving in the potential well (compared to ). The first three normalized supersymmetric eigenstates are given by , ; , ; , .
In particle physics, supersymmetry has been proposed as a connection between bosons and fermions. Although this is a beautiful theory, there is, as yet, no experimental evidence that Nature contains supersymmetry. If it does exist, it must be a massively broken symmetry. It is possible that the Large Hadron Collider will find supersymmetric partners of some known particles.
  • Contributed by: S. M. Blinder
  • With suggestions by Jeremy Michelson

SNAPSHOTS

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

DETAILS

Reference: A. Khare, "Supersymmetry in Quantum Mechanics," arXiv, 2004.
    • 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+