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Journal of physics. A, Proceedings of the Physical Society. General

Lower bounds to second-order corrections to eigenvalues observables of stationary quantum systems

C. S. Sharma

Published Jan 1, 1968• 2 citations
Publisher

Abstract

This letter finds lower bounds to the second-order correction to the eigenvalue of an observable for the ith excited state of a stationary quantum system.
DOI
10.1088/0305-4470/2/1/020

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The expectation value of the Hamiltonian H for a many-electron system may be expressed in terms of the reduced second-order density matrix and the “reduced” Hamiltonian K for a two-electron system. It may be shown that the antisymmetric component HAS of the original Hamiltonian is an “outer projection” of K with respect to the antisymmetrizer O, so that HAS = OKO. This result implies that the eigenvalues of K are, in order, lower bounds to the eigenvalues of HAS, i.e. to the eigenvalues of H associated with the antisymmetry requirement. The expectation values of K are hence often below the ground state energy of H, and the eigenvalues may be used for the calculation of lower bounds. Some numerical applications to atoms serve as an illustration, and it is shown that the lower bounds become worse as the number of electrons increases. The implications for the “representability problem” are discussed.