Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states

Abstract

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the . The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated with another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models that have previously appeared in the chemical physics literature.

Koopman wavefunctions and classical–quantum correlation dynamics

“…z =´zρ d 6 z =´Ψ * Z − Ψ d 6 z. As shown in [6] by adopting the harmonic oscillator gauge (8), this last relation returns the usual Ehrenfest equations for the expectation dynamics of canonical observables.…”

Moyal Bracket and Ehrenfest’s Theorem in Born–Jordan Quantization

“…Equation 5is characteristic of BJ quantization as shown in [3]. We emphasize that this fact is not related in any way to Groenewold's and van Hove's result because the latter does not preclude quantizations satisfying Equation (5).…”

“…We also mention that Bonet-Luz and Tronci have studied, in [5], Ehrenfest expectation values from a dynamical and geometric point of view focusing on Gaussian states. It would certainly be interesting to develop these techniques using the results in the present paper.…”

Geometry and dynamics of Gaussian wave packets and their Wigner transforms

“…Recently, Bonet-Luz and Tronci [5] discovered a non-canonical Poisson bracket that describes the dynamics of the Gaussian Wigner function 1…”