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Journal of Physics A Mathematical and General

Lower bounds to the overlap

Thomas Hoffmann-Ostenhof, Maria Hoffmann-Ostenhof, Gottfried Olbrich

Published Jan 1, 1976• 6 citations
Publisher

Abstract

A variational lower bound to the overlap of the ground state psi 0 of a quantum mechanical system with an approximate wavefunction phi is given. A simplified non-variational formula is deduced from this lower bound to mod ( phi mod psi 0 ) mod . Both versions are illustrated with an application to the hydrogen atom and are compared with the lower bounds of Eckart and Wang. The non-variational version turns out to be a useful alternative to Eckart's bound.

DOI
10.1088/0305-4470/9/1/007

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Journal of Mathematical Physics•May 1, 1970•4 citations
A variational lower bound to the overlap of the ground state psi 0 of a quantum mechanical system with an approximate wavefunction phi is given. A simplified non-variational formula is deduced from this lower bound to mod ( phi mod psi 0 ) mod . Both versions are illustrated with an application to the hydrogen atom and are compared with the lower bounds of Eckart and Wang. The non-variational version turns out to be a useful alternative to Eckart's bound.
A general formalism is presented which furnishes upper and lower variational bounds on squared overlaps mod ( theta , chi ) mod 2 , where chi and theta are vectors in a Hilbert space. The bounding functionals involve a self-adjoint operator A and a second operator B, such that A theta =0 and B chi =b chi . Most of the non-variational bounds derived by other workers on quantum-mechanical overlaps emerge from the formalism, together with optimal variational improvements in every case. In addition many new bounds are obtained.
The accuracy with which a trial function φ approximates the true wavefunction ψ is quantitatively assessed by the overlap integral S = 〈φ | ψ〉. Upper and lower bounds to S therefore furnish direct criteria of accuracy of the approximation φ and also of the associated physical properties. The available literature on overlap estimates is assembled and critically discussed from a unified point of view, based upon a method of determinantal inequalities. In particular, the relationships among the various approaches are pointed up, several results are extended or generalized, and some new results are obtained. Finally, the various formulas are illustrated by numerical applications to some simple soluble problems.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
An intrinsic measure of the quality of a variational wave function is given by its overlap with the ground state of the system. We derive a general formula to compute this overlap when quantum dynamics in imaginary time is accessible. The overlap is simply related to the area under the E(tau) curve, i.e., the energy as a function of imaginary time. This has important applications to, for example, quantum Monte Carlo simulations where the overlap becomes as a simple by-product of routine simulations. As a result, we find that the practical definition of a good variational wave function for quantum Monte Carlo simulations, i.e., fast convergence to the ground state, is equivalent to a good overlap with the actual ground state of the system.