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International Journal of Modern Physics B

On the Uniform Upper Bounds to the Ground State Energy of Quantum System

Alexey N. Kireev

Published Apr 20, 1997• 0 citations
Publisher

Abstract

We derive a set of improving uniform upper bounds to the ground state energy of a quantum system, which provides a natural generalization of the Ritz variational principle. The bounds have a general character, do not depend on the structure of Hamiltonian of a quantum system and converge to its exact ground state energy. As an illustration of the method proposed, we consider a simple example of the shifted harmonic oscillator.

DOI
10.1142/s0217979297000629

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We derive a set of improving uniform upper bounds to the ground state energy of a quantum system, which provides a natural generalization of the Ritz variational principle. The bounds have a general character, do not depend on the structure of Hamiltonian of a quantum system and converge to its exact ground state energy. As an illustration of the method proposed, we consider a simple example of the shifted harmonic oscillator.
The variational principle is revisited in the context of finding the upper and lower bounds to the ground state energy. It is shown how the variational principle can be employed to find the lower bound by partitioning the Hamiltonian into several parts. We demonstrate how the variational principle can be used to find the exact ground states in some special cases. We consider the harmonic oscillator and the Ising Hamiltonian on a bipartite lattice which are familiar yet instructive examples for students.
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.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. ƒree time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. ƒe WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. ƒe third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. Three time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. The WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. The third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.