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

Lagrange multipliers and bounds to quantum mechanical properties

M. Cohen, Tova Feldmann

Published Nov 1, 1971• 5 citations
Publisher
DOI
10.1088/0305-4470/4/6/003

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Abstract We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.
We investigate the Lagrange-multiplier method which is frequently used in strongly correlated-electron systems to impose the constraints. We assume that the constraint operators commute with the Hamiltonian. It may be implicitly believed that the Lagrange multipliers are determined uniquely due to the variational conditions. We discuss whether this is true or not by using the exact diagonalization method. For the mean-field Hamiltonian which does not commute with the constraint operators, the Lagrange multipliers may be determined uniquely.