Theory of boundary perturbation and the compressed hydrogen molecular ion

Abstract

Fröhlich's theory of boundary perturbation may be used for the treatment of bounded quantum mechanical systems of large extent but it suffers from the drawback that on the bounding wall derivatives of the exact, perturbed wave functions must be assumed to be approximately equal to those of the corresponding unperturbed wave functions, which is seldom the case. In this paper a simple method is devised for determining the derivatives in question in terms of the unperturbed wave functions. For problems like the harmonic oscillator or the hydrogen atom, this method leads at once to a general asymptotic formula for the eigenvalues previously derived by Hull and Julius. The method is also applied to the somewhat more complex problem of the hydrogen molecular ion in a spheroidal box and the increase of the total energy and the increase of the kinetic energy of the ion are calculated as functions of pressure upto 15,000 atmospheres. A comparison with the results derived earlier by Cottrell by the variational method shows large discrepancies and hence, in order to obtain a check on the results derived by the perturbation method, variational calculations are repeated using a trial function more suitable than the one employed by Cottrell. The energies thus obtained are in excellent agreement with those derived by perturbation theory.