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

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