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Zeitschrift für Naturforschung A

Modification of the Delves Variation Principle

K. Robert Müller

Published Jan 1, 1965• 0 citations
Publisher

Abstract

A variation principle, similar to the Delves principle, is given for the (real) wave function itself. The principle does not give a bound. The relationship between both principles is discussed. — After proper generalization a variation principle for arbitrary probability amplitudes (or generalized Fourier coefficients) is obtained.
DOI
10.1515/zna-1965-0104

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Variational principles are designed for the solution of the Schr\"odinger equation when a point source is placed in the presence of an inhomogeneous, absorbing medium represented by an arbitrary complex potential function. When the point source is allowed to recede to infinity, these stationary structures reduce to variational principles for the wave function in the standard scattering problem, namely the outgoing solution to the Schr\"odinger equation for an incident plane wave. Finally in the asymptotic region, the well-known bifunctional variational principles for the transition amplitudes arise automatically from the stationary forms for the wave function describing the standard scattering problem. A few examples leading to variationally improved wave functions are discussed.
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A new variational principle for the energy eigenvalues of a confined quantum system is presented. Whereas the exact wave function \ensuremath{\Psi} must vanish on the bounding surface of the region, the trial function \ensuremath{\psi} in this principle need not obey any specific boundary condition. Calculationally, the method is similar to the conventional variational method except that kinetic energy turns out to be a weighted average of -F\ensuremath{\psi}${\mathrm{\ensuremath{\nabla}}}^{2}$\ensuremath{\psi} d\ensuremath{\tau} and F(\ensuremath{\nabla}\ensuremath{\psi})\ensuremath{\cdot}(\ensuremath{\nabla}\ensuremath{\psi})d\ensuremath{\tau} in the ratio 2:-1. Although the principle is not a definite (minimum) one, good results are obtained in several examples.