Accuracy of Approximate Eigenstates

TL;DR

This paper evaluates the accuracy of approximate eigenstates obtained via variational methods, proposing new criteria based on operator commutators, and applies these to the relativistic spinless Salpeter equation in quantum physics.

Contribution

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

Findings

Proposes commutator-based criteria for eigenstate accuracy.

Applies criteria to the relativistic Salpeter equation.

Provides insights into the precision of variational approximations.

Abstract

Besides perturbation theory, which requires, of course, 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 some variational method, are proposed here as new criteria for the accuracy of variational eigenstates. These considerations are applied to that Hamiltonian the eigenvalue problem of which defines the "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.