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International Journal of Nonlinear Sciences and Numerical Simulation

Application of the Lagrange Multiplier Method the Semi-Inverse Method to the Search for Generalized Variational Principle in Quantum Mechanics

Hao Tian-hu

Published Jan 1, 2003• 25 citations
Publisher
DOI
10.1515/ijnsns.2003.4.3.311

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For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.