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Journal of Physics A Mathematical Nuclear and General

A generalization of Weinstein's procedure for calculating bounds to quantum-mechanical quantities

M. Cohen, Tova Feldmann

Published Mar 21, 1974• 3 citations
Publisher

Abstract

A simple unified procedure, based on a generalization of a method first used by Weinstein in 1934, yields effective lower and upper bounds to energy levels and overlap integrals of quantum-mechanical systems. A further generalization yields bounds to expectation values.
DOI
10.1088/0305-4470/7/5/006

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arXiv:2208.09370•Progress of Theoretical and Experimental Physics•Jan 5, 2023•11 citations
Weinstein's modification of the Ritz principle is used to derive (1) a lower bound for the nnn-th energy-level of a quantum mechanical system if a lower bound for the $(n+1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known; and (2) an upper bound for the nnn-th level if an upper bound for the $(n\ensuremath{-}1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known.
Abstract The range of motion of a particle with certain energy E confined in a potential is determined from the energy conservation law in classical mechanics. The counterpart of this question in quantum mechanics can be regarded as what is the possible range of expectation values of the position operator 〈x〉 of a particle that satisfies E = 〈H〉. This range depends on the state of the particle, but the universal upper and lower bounds, which are independent of the state, must exist. In this study, we show that these bounds can be derived by using the bootstrap method. We also point out that the bootstrap method can be regarded as a generalization of the uncertainty relations, meaning that the bounds are determined by the uncertainty relations in a broad sense. Furthermore, the bounds on possible expectation values of various quantities other than position can be determined in the same way. However, in the case of multiple identical particles (bosons and fermions), we find some difficulty in the bootstrap method. Because of this issue, the predictive power of the bootstrap method in multi-particle systems is limited in the derivation of observables including energy eigenstates. In addition, we argue an application of the bootstrap method to thermal equilibrium states. We find serious issues that temperature and entropy cannot be handled. Although we have these issues, we can derive some quantities in micro-canonical ensembles of integrable systems governed by generalized Gibbs ensembles.