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.