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European Journal of Physics

Reflection and non-reflection of particle wavepackets

Timothy Cox, John Lekner

Published May 19, 2008• 12 citations
Publisher

Abstract

Exact closed-form solutions of the time-dependent Schrödinger equation are obtained, describing the propagation of wavepackets in the neighbourhood of a potential. Examples given include zero reflection, total reflection and partial reflection of the wavepacket, for the , 1/ x 2 and δ( x ) potentials, respectively. In the first two of these cases the results are obtained using the methods of elementary supersymmetric quantum mechanics. This gives an introduction to supersymmetry in its simplest form, suitable for graduate or advanced undergraduate students. Animations of wavepacket propagation are provided.

DOI
10.1088/0143-0807/29/4/002

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Exact closed-form solutions of the time-dependent Schrödinger equation are obtained, describing the propagation of wavepackets in the neighbourhood of a potential. Examples given include zero reflection, total reflection and partial reflection of the wavepacket, for the , 1/ x 2 and δ( x ) potentials, respectively. In the first two of these cases the results are obtained using the methods of elementary supersymmetric quantum mechanics. This gives an introduction to supersymmetry in its simplest form, suitable for graduate or advanced undergraduate students. Animations of wavepacket propagation are provided.
Abstract Here we report a project in which time-dependent supersymmetry has been employed to derive a new potential and eigenfunctions that satisfy the Schrödinger equation. The supersymmetry method is outlined and we apply it to a wavefunction obeying the free-particle Schrödinger equation. This leads to an exactly soluble model in which a quantum particle is seen to ‘surf’ on a time-dependent potential. The model can be solved and understood within both classical and quantum mechanics and the relationship between the two approaches is discussed. The mathematics of this formalism is accessible to a final year British undergraduate making supersymmetry derived Hamiltonians suitable as a final year theoretical physics research project.
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