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Journal of Anhui Institute of Architecture & Industry

POISSON'S BRACKET IN ANALYTICAL MECHANICS AND QUANTUM MECHANICS

De Zhu

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

Poisson's bracket is analyzed and its major function is summarized in two different mechanics systems respectively.The relationship between quantum mechanics and analytical mechanics is discussed and their unity is clarified.

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We introduce the Poisson bracket operator, which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation, which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to ccc-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore, the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to nonstandard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where ccc-number and qqq-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.
The widely accepted approach to the foundation of quantum mechanics is that the Poisson bracket, governing the non-commutative algebra of operators, is taken as a postulate with no underlying physics. In this manuscript, it is shown that this postulation is in fact unnecessary and may be replaced by a few deeper concepts, which ultimately lead to the derivation of Poisson bracket. One would only need to use Fourier transform pairs and Kramers-Kronig identities in the complex domain. We present a definition of Hermitian time-operator and discuss some of its basic properties.
We describe an ppp-mechanical (see funct-an/9405002 and quant-ph/9610016) brackets which generate quantum (commutator) and classic (Poisson) brackets in corresponding representations of the Heisenberg group. We \emph{do not} use any kind of semiclassic approximation or limiting procedures for ℏ→0\hbar \to 0ℏ→0. Harmonic oscillator considered within the approach. Keywords: Classic and quantum mechanics, Hamilton and Heisenberg equations, Poisson brackets, commutator, Heisenberg group.