arXiv:quant-ph/0604076

Evolution Law of Quantum Observables from Classical Hamiltonian in Non-Commutative Phase Space

Daniela Dragoman

Published Apr 11, 2006• 1 citation

Abstract

The evolution equations of quantum observables are derived from the classical Hamiltonian equations of motion with the only additional assumption that the phase space is non-commutative. The demonstration of the quantum evolution laws is quite general; it does not rely on any assumption on the operator nature of x and p and is independent of the quantum mechanical formalism.

DOI: 10.48550/arxiv.quant-ph/0604076

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arXiv:quant-ph/0604076

Evolution Law of Quantum Observables from Classical Hamiltonian in Non-Commutative Phase Space

Daniela Dragoman

Published Apr 11, 2006• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/0604076

Related papers

Matrix Algebras in Non-Hermitian Quantum Mechanics

Alessandro Sergi

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.

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arXiv:1402.0396European Journal of PhysicsMay 9, 20144 citations

F=\frac{dP}{dt}

P=m\frac{dV}{dt}

, and

\left[X,P\right]=i

. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.