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Physical Review A

Exact perturbation theory for quantum-mechanical systems within boxes

Francisco Marcelo Fernández, Eduardo Alberto Castro

Published Dec 1, 1992• 8 citations
Publisher

Abstract

We apply perturbation theory to the Schr\"odinger equation for a system caged in a one-dimensional box with impenetrable walls and show how to obtain the perturbation corrections to both the energy eigenvalues and eigenfunctions in a systematic exact way when the potential function within the box is a polynomial. As an illustrative example we consider a linear potential obtaining analytic expressions for the perturbation corrections in terms of the quantum number.

DOI
10.1103/physreva.46.7288

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We apply perturbation theory to the Schr"odinger equation for a system caged in a one-dimensional box with impenetrable walls and show how to obtain the perturbation corrections to both the energy eigenvalues and eigenfunctions in a systematic exact way when the potential function within the box is a polynomial. As an illustrative example we consider a linear potential obtaining analytic expressions for the perturbation corrections in terms of the quantum number.
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Rayleigh–Schrödinger perturbation theory is used to find energy and wavefunction corrections for a particle confined in a box with infinite walls. The resulting perturbation equations have simple, analytic solutions. The procedure for normalization of the wavefunction corrections is demonstrated, and the perturbation energies and normalized wavefunctions are shown to approach the exact values. The method is applied to both the ground state and first excited state and can become a valuable component of a student’s introduction to perturbation analysis.
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