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Journal of Mathematical Physics

Bound state variational principle employing a discontinuous trial function

Kenneth R. Brownstein

Published Jan 1, 1995• 4 citations
Publisher

Abstract

A new variational principle for the bound state energy eigenvalues of a quantum system is presented. In it, not only ∇ψ(r) but also ψ(r) itself can be discontinuous across a surface. That is, the trial ψ(r) can have both a kink and a jump across a surface. This allows the use of different basis functions in different regions of space without having to impose any matching conditions across the surface. All parameters appearing in the principle can be variational parameters; none are needed to help invoke any continuity conditions on the trial function. Although the principle is not a definite (minimum) one, good results are obtained in several examples.

DOI
10.1063/1.531327

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A new variational principle for the bound state energy eigenvalues of a quantum system is presented. In it, not only ∇ψ(r) but also ψ(r) itself can be discontinuous across a surface. That is, the trial ψ(r) can have both a kink and a jump across a surface. This allows the use of different basis functions in different regions of space without having to impose any matching conditions across the surface. All parameters appearing in the principle can be variational parameters; none are needed to help invoke any continuity conditions on the trial function. Although the principle is not a definite (minimum) one, good results are obtained in several examples.
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.
Recent work [Phys. Rev. A 9, 108 (1974)] on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Phys. Rev. A 9, 108 (1974) originally were based.
In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.
A more reasonable trial ground state wave function is constructed for the relative motion of an interacting two-fermion system in a one-dimensional (1D) harmonic potential. At the boundaries both the wave function and its first derivative are continuous and the quasi-momentum is determined by a more practical constraint condition which associates two variational parameters. The upper bound of the ground state energy is obtained by applying the variational principle to the expectation value of the Hamiltonian of relative motion on the trial wave function. The resulted energy and wave function show better agreement with the analytical solution than the original proposal.
The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.