Quality of Variational Trial States

TL;DR

This paper evaluates the accuracy of variational trial states in quantum eigenvalue problems, proposing new criteria based on commutator matrix elements, with applications to the relativistic spinless Salpeter equation.

Contribution

It introduces novel measures for the accuracy of variational eigenstates using commutator matrix elements, specifically applied to the relativistic Salpeter equation.

Findings

New criteria for variational eigenstate accuracy are proposed.

Application to the spinless Salpeter equation demonstrates the criteria's effectiveness.

Insights into the approximation quality of variational methods for relativistic quantum systems.

Abstract

Besides perturbation theory (which clearly requires the knowledge of the exact unperturbed solution), variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H usually can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of the approximate eigenstates derived by variational techniques are scrutinized. In particular, the matrix elements of the commutator of the operator H and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by variational methods are proposed as new criteria for the accuracy of variational eigenstates. These considerations are applied to precisely that Hamiltonian for which the eigenvalue problem defines the well-known spinless Salpeter equation. This bound-state wave equation may be regarded as (the most straightforward) relativistic generalization of the usual nonrelativistic Schroedinger formalism, and is frequently used to describe, e.g., spin-averaged mass spectra of bound states of quarks.