Abstract
The usual Poisson bracket { A , B } can be identified with the so-called Moyal bracket { A , B } M for larger classes of symbols than was previously thought, provided that one uses the Born–Jordan quantization rule instead of the better known Weyl correspondence. We apply our results to a generalized version of Ehrenfest’s theorem on the time evolution of averages of operators.
Highlights
IntroductionA famous theorem from quantum statistical mechanics says that the evolution of the quantum averages with respect to a state ρ of a quantum observable A under a Hamiltonian evolution obeys the generalized Ehrenfest equation (Messiah [1])
A famous theorem from quantum statistical mechanics says that the evolution of the quantum averages with respect to a state ρ of a quantum observable A under a Hamiltonian evolution obeys the generalized Ehrenfest equation (Messiah [1]) d dt A =1 ih [A, H] . (1)If we were able to write the commutator as a quantization of the Poisson bracket {A, H} of the classical observables corresponding to A and H by the Weyl correspondence, we could rewriteEquation (1) as d dt ={A, H}(x, p)ρ(x, p, t)dpdx (2)where ρ(x, p, t) is the Wigner distribution of the state ρ at time t
1 2 z this is “Weyl’s product rule”. It is customary in quantum mechanical texts to say that the Weyl symbol c is the “Moyal product” [6] of A and B and to write c = A h B
Summary
A famous theorem from quantum statistical mechanics says that the evolution of the quantum averages with respect to a state ρ of a quantum observable A under a Hamiltonian evolution obeys the generalized Ehrenfest equation (Messiah [1]). Where {A, H}M is the so-called “Moyal bracket” It was only recently recognized [2,3] that the Dirac correspondence, Equation (3), holds for a large class of observables provided that we use the Born–Jordan (BJ) quantization scheme instead of the usual Weyl quantization (this was already noticed but not fully developed by Kauffmann [4] a few years ago). The main result we will prove in this paper is the following (Proposition 6): Let ρ(z, t) be the Wigner distribution at time t and let A be a quantum observable obtained by any quantization procedure from a classical observable (symbol) of the type A(x, p) = S(x) + V(p) with S and V smooth functions of polynomial growth. It would certainly be interesting to develop these techniques using the results in the present paper
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