Poisson bracket operator
- URL: http://arxiv.org/abs/2104.11780v2
- Date: Mon, 4 Oct 2021 15:42:33 GMT
- Title: Poisson bracket operator
- Authors: T. Koide
- Abstract summary: We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket.
We derive the quantum canonical equation which describes the time evolution of operators.
The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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
c-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 non-standard system where the
Heisenberg equation is not defined. As an example, we consider the application
to the system where a c-number and a q-number particles coexist. The derived
dynamics satisfies the Ehrenfest theorem and the energy and momentum
conservations.
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