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CRC Press eBooks

Equations of Motion for Operators

Gordon Baym

Published Mar 5, 2018• 0 citations
Publisher

Abstract

The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.

DOI
10.1201/9780429499265-5

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The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
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This chapter attempts to reconnect quantum mechanics to classical mechanics by comparing the role of the Poisson bracket used in classical mechanics to the commutator used in quantum mechanics. After introducing Planck’s quantization method, the operators for position and linear momentum are defined, their eigenvalues and eigenvectors are derived, and their mutual commutator is taken and compared to the analogous result in classical mechanics. Finally, the Heisenberg’s uncertainty principle is introduced, and its implications briefly discussed.