Variational principles for particles and fields in Heisenberg matrix mechanics

Journal Article · Wed Oct 01 00:00:00 EDT 1980 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.524359· OSTI ID:6895321

Klein, A; Li, C T; Vassanji, M

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.

• Research Organization: Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
• DOE Contract Number: EY-76-C-02-3071
• OSTI ID: 6895321
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 21:10; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BANACH SPACE
COMMUTATION RELATIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
EQUATIONS OF MOTION
FIELD THEORIES
HAMILTONIANS
HEISENBERG PICTURE
HILBERT SPACE
MANY-BODY PROBLEM
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATRIX ELEMENTS
MECHANICS
ONE-DIMENSIONAL CALCULATIONS
QUANTUM MECHANICS
QUANTUM OPERATORS
SPACE
VARIATIONAL METHODS