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arXiv:2403.12974 · Physics Essays

Nonseparable wave evolution equations in quantum kinetics

Christos Dedes

Published Sep 18, 2024• 0 citations
arXivHTMLPDF

Abstract

A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton‐Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density.

DOI
10.4006/0836-1398-37.3.215

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A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton‐Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density.
A phase space description of Schrödinger dynamics is provided in terms of a quantum kinetic formalism relying on the introduction of an appropriate extension of the well-known Wigner transform, also accounting for time delocalizations. This “space-time Wigner distribution,” built up in the framework of two-time correlation functions, is shown to be governed by a non-Markovian, integro-differential equation of convolution type. Its utility in investigating long time dynamics of quantum systems is also discussed and illustrated with some examples.
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Abstract Recently, a hydrodynamic description of local equilibrium dynamics in quantum integrable systems was discovered. In the diffusionless limit, this is equivalent to a certain ‘Bethe–Boltzmann’ kinetic equation, which has the form of an integro-differential conservation law in ( 1 + 1 ) D. The purpose of the present work is to investigate the sense in which the Bethe–Boltzmann equation defines an ‘integrable kinetic equation’. To this end, we study a class of N dimensional systems of evolution equations that arise naturally as finite-dimensional approximations to the Bethe–Boltzmann equation. We obtain non-local Poisson brackets and Hamiltonian densities for these equations and derive an infinite family of first integrals, parameterized by N functional degrees of freedom. We find that the conserved charges arising from quantum integrability map to Casimir invariants of the hydrodynamic bracket and their group velocities map to Hamiltonian flows. Some results from the finite-dimensional setting extend to the underlying integro-differential equation, providing evidence for its integrability in the hydrodynamic sense.
We follow and modify the Feshbach–Villars formalism by separating the Klein–Gordon equation into two coupled time-dependent Schrödinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.
The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.
A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.