December 15, 2016

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.