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American Journal of Physics

Approximate energy levels of bound quantum systems

Francisco Marcelo Fernández, Eduardo Alberto Castro

Published May 1, 1984• 5 citations
Publisher

Abstract

A simple approximate method based on variational principles is used to obtain energy eigenvalues for bound quantum systems. It is shown that the approximate eigenvalues satisfy both virial and Hellmann–Feynman theorems. Results obtained by this method are compared with the exact solutions of the Schrödinger equation.
DOI
10.1119/1.13630

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In most cases solving a quantum mechanical problem exactly is not always possible. In such a case, an analytic solution to the Schrödinger equation is appropriate. This results to numerical solutions that may not necessarily be the best approach. Interest sometimes may not necessarily be in the detailed eigenfunctions, but rather in the energy levels and the qualitative features of the eigenfunctions. So, the numerical solutions are usually less intuitively reasonable. It is possible to find the values of the energy levels to be slightly sensitive to the deviation of the wave function from its true form. So, the expectation value of the energy for an approximate wave function can be a very good estimate of the corresponding energy eigenvalue. Using an approximate wave function dependent on some set of variational parameters and minimizing its energy by these variational parameters yield such energy estimates. This technique is the so-called variational method due to this minimization procedure.