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Europhysics Letters (EPL)

Wave propagation and “Landau-type” damping from Bohm potential

D. Mostacci, Vincenzo G. Molinari, Francesco Pizzio

Published Mar 26, 2008• 3 citations
Publisher

Abstract

From Bohm's interpretation of quantum mechanics, a quantum kinetic equation (QKE) can be derived. It is found that waves propagate in force-free gases of non interacting particles, only due to Bohm potential. In the present article the existence of Landau damping in such propagations is investigated. It is found that the Bohm potential alone gives indeed rise to a damping entirely analogous to the classical Landau damping, both for bosons and for weakly degenerate fermions.

DOI
10.1209/0295-5075/82/20002

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From Bohm's interpretation of quantum mechanics, a quantum kinetic equation (QKE) can be derived. It is found that waves propagate in force-free gases of non interacting particles, only due to Bohm potential. In the present article the existence of Landau damping in such propagations is investigated. It is found that the Bohm potential alone gives indeed rise to a damping entirely analogous to the classical Landau damping, both for bosons and for weakly degenerate fermions.
Abstract Bohm’s interpretation of Quantum Mechanics leads to the derivation of a Quantum Kinetic Equation (QKE): in the present work, propagation of waves in charged quantum gases is investigated starting from this QKE. Dispersion relations are derived for fully and weakly degenerate fermions and bosons (for the latter above critical temperature) and the differences discussed. Use of a kinetic equation permits investigation of “Landau-type” damping: it is found that the presence of damping in fermion gases is dependent upon the degree of degeneracy, whereas it is always present in boson gases. In fully degenerate fermions a phenomenon appears that is akin to the “zero sound” propagation.
Starting from Bohm’s point of view on Quantum Mechanics, adopting the Bohm potential description, a Quantum Kinetic Equation can be derived. When molecular interactions are described as a self-consistent field, a Quantum Vlasov Equation (QVE) can be written. With this QVE, the evolution of a small perturbation can be described, yielding a wave propagation with or without a damping coefficient, depending on conditions. With this approach applied to a liquid of either fermions or bosons, the propagation of waves in quantum liquids is described.
In Bohm's interpretation of Quantum Mechanics, quantum effects are governed by a “quantum potential” (known as Bohm potential), and particles follow definite trajectories. In the present work, Liouville's theorem is invoked, an appropriate Liouville equation is derived, and following the BBGKY method a quantum kinetic equation (QKE) is derived. To demonstrate the working of the QKE, two examples of application are presented: the thermal equilibrium of a quantum gas and the propagation of disturbances in a force free gas of non-interacting bosons. In contrast to the classical collisionless Boltzmann equation, waves are found to be possible in the absence of interaction or external forces, due only to Bohm potential (zero sound propagation).
Electromagnetic interactions of a single charged particle in the presence of damping are investigated theoretically in this work. Classically, by considering the explicit example of a particle in a constant magnetic field, it is shown that the minimal coupling prescription is inadequate for the infinitely large number of Hamiltonians from which the equations of motion in the presence of damping are derivable by Hamilton's equations. The use of minimal coupling procedure to switch on electromagnetic interactions in phenomenological quantum equations of damped motion is shown to lead to incorrect equations in the classical limit. It is necessary to consider the microscopic models of damping to allow for electromagnetic interactions in these systems. Two microscopic models for damping involving a linear coupling with a heat bath of oscillators are considered in detail and it is shown that correct Heisenberg equations of motion follow upon elimination of bath degrees of motion so that damping and Lorentz force do not affect each other. However, the field couples to the mechanical momentum as mv-(e/c)A rather than to the canonical momentum. The linearly coupled quantal case is analogous to the classical situation. The Schr\"odinger-Langevin equation contains an extra term Fr{\mathcal{F}}^{\mathrm{r}}Fr A\ensuremath{\cdot}dr'; however, the gauge invariance character and the Aharonov-Bohm effect are unaffected. Two nonlinear models of damping are studied by considering the exchange of bosons in the different states of the central particle. In this situation the Lorentz force is not recovered in the classical limit.
A semiclassical method is used to study Landau damping of transverse pseudo-spin waves in harmonically trapped ultracold gases in the collisionless Boltzmann limit. In this approach, the time evolution of a spin is calculated numerically as it travels in a classical orbit through a spatially dependent mean field. This method reproduces the Landau damping results for spin-waves in unbounded systems obtained with a dielectric formalism. In trapped systems, the simulations indicate that Landan damping occurs for a given spin-wave mode because of resonant phase space trajectories in which spins are "kicked out" of the mode (in spin space). A perturbative analysis of the resonant and nearly resonant trajectories gives the Landau damping rate, which is calculated for the dipole and quadrupole modes as a function of the interaction strength. The results are compared to a numerical solution of the kinetic equation by Nikuni et al.