aqora

Heisenberg, matrix mechanics, and the uncertainty principle

S. Lakshmibala

ResonanceAug 1, 20044 citations

Heisenberg evolution of matrix product operators by time-dependent variational principle

Xiangyu Cao, Ehud Altman

Bulletin of the American Physical SocietyMar 8, 20180 citations

Variational principles for particles and fields in Heisenberg matrix mechanics

Abraham Klein, Ching‐Teh Li, Moyez G. Vassanji

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.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.

Journal of Mathematical PhysicsOct 1, 198013 citations

Heisenberg’s Matrix Mechanics

Alexander Komech

Springer eBooksOct 23, 20120 citations

Invariant Tori and Heisenberg Matrix Mechanics: A New Window on the Quantum- Classical Correspondence

William R. Greenberg, Abraham Klein, Ching‐Teh Li

Despite the seminal connection between classical multiply periodic motion and Heisenberg matrix mechanics, we show that there are fundamental, previously undisclosed aspects of this quantum-classical correspondence. These include a quantum variational principle that implies the classical variational principle for invariant tori and the connection between commutation relations and quantization of action variables. Possible applications are described briefly.

Physical Review LettersAug 14, 199527 citations

WKB approximation for bound states by Heisenberg matrix mechanics

Abraham Klein

The WKB approximation for bound states is derived from Heisenberg matrix mechanics. It is shown that the result can be obtained from the Bohr–Sommerfeld quantization of a suitably chosen solution of the classical equations of motion. It is also derived from the commutation relation and from a quantum form of Hamilton’s variational principle.

Journal of Mathematical PhysicsJan 1, 197816 citations

Variational principles and Heisenberg matrix mechanics

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