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Scientific Reports

Dirac potential in a rotational dissipative quantum system

Yi-Rong Ma, Wei Jia, Shirong Lin, Qing Bo Zhao

Published Feb 7, 2019• 0 citations
Publisher

Abstract

This study proposes the usage of an effective potential to investigate a dissipative quantum system with rotational velocity. After gauge transformation, a Doebner- Goldin equation (DGE) that describes the dissipative quantum system with a Dirac potential is obtained. The DGE is solved based on constraint of vertical relation between the rotational velocity field and density gradient when a harmonic oscillator model is considered. It is observed that the dissipative quantum system is directly equivalent to a monopole system and that the two gauge potentials that are given by Wu and Yang in the north and south hemispheres can be reproduced. Furthermore, a set of gauge-invariant parameters is obtained to discuss the dissipation characteristics of the system.

DOI
10.1038/s41598-018-35763-z

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This study proposes the usage of an effective potential to investigate a dissipative quantum system with rotational velocity. After gauge transformation, a Doebner- Goldin equation (DGE) that describes the dissipative quantum system with a Dirac potential is obtained. The DGE is solved based on constraint of vertical relation between the rotational velocity field and density gradient when a harmonic oscillator model is considered. It is observed that the dissipative quantum system is directly equivalent to a monopole system and that the two gauge potentials that are given by Wu and Yang in the north and south hemispheres can be reproduced. Furthermore, a set of gauge-invariant parameters is obtained to discuss the dissipation characteristics of the system.
A realistic analysis shows that constraining a quantomechanical system produces the effective dynamics to be coupled with abelian/nonabelian gauge fields and quantum potentials induced by the intrinsic and extrinsic geometrical properties of the constraint’s surface. This phenomenon is observable in the effective rotational motion of some simple polyatomic molecules. By considering specific examples it is shown that the effective Hamiltonians for the nuclear rotation of linear and symmetric top molecules are equivalent to that of a charged system moving in a background magnetic-monopole field. For spherical top molecules an explicit analytical expression of a non-abelian monopole-like field is found. Quantum potentials are also relevant for One of the first questions addressed by researchers who first studied the properties of surfaces embedded in the three-dimensional euclidean space, at
Recently, it is shown that the extended phase space formulation of quantum mechanics is a suitable technique for studying the quantum dissipative systems. Here, as a further application of this formalism, we consider a dissipative system of charged particles interacting with an external time dependent electric field. Such a system has been investigated by Buch and Denman, and two distinct solutions with completely different structure have been obtained for Schrödinger's equation in two different gauges. However, by generalizing the gauge transformations to the phase space and using the extended phase space technique to study the same system, we demonstrate how both gauges lead to the same conductivity, suggesting the recovery of gauge invariance for this physical quantity within the extended phase space approach.
In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.
Abstract We consider the Dirac equations in polar form proving that they can equivalently be re-configured into a system of equations consisting of derivatives of the velocity density plus the Hamilton–Jacobi equation, giving the momentum in terms of relativistic quantum potentials (i.e. displaying first-order derivatives of the two degrees of freedom of the spinor field): this system is said to have Madelung structure. Conservation laws, second-order equations and multi-valuedness are also discussed.