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Physical Review E

Space fractional Wigner equation and its semiclassical limit

Benjamin A. Stickler, E. Schachinger

Published Dec 15, 2011• 4 citations
Publisher

Abstract

Manifestations of space fractional quantum mechanics (SFQM), as it was formulated by Laskin [Phys. Rev. E 62, 3135 (2000)], are deemed to offer a better physical interpretation of L\'evy flight statistics on a quantum mechanical level. We start with the SFQM Schr\"odinger equation characterized by a L\'evy flight index $\ensuremath{\alpha}\ensuremath{\in}(1,2)$, perform a Wigner transform, and draw the limit $\frac{\ensuremath{\hbar}}{E\ensuremath{\tau}}\ensuremath{\rightarrow}0$ (i.e., let the observed energy scale $E$ go to infinity in comparison to the quantization given by $\ensuremath{\hbar}/\ensuremath{\tau}$). In order to obtain classical transport equations two possible substitutions for the terms ${|p|}^{\ensuremath{\alpha}}$ and $|{p}^{\ensuremath{'}}{|}^{\ensuremath{\alpha}}$ which appear in von Neumann's equation are presented. It is demonstrated that they conform to the criteria for a successful Wigner transform. Their benefits and caveats are discussed in detail. We find, that, indeed, SFQM manifests itself in an anomalous kinetic term of the free particle's motion and, assuming an external potential diagonal in momentum space for the sake of simplicity, in corresponding anomalous terms in the resulting drift current. All our results reduce to the classical forms in the limit $\ensuremath{\alpha}=2$.

DOI
10.1103/physreve.84.061129

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Manifestations of space fractional quantum mechanics (SFQM), as it was formulated by Laskin [Phys. Rev. E 62, 3135 (2000)], are deemed to offer a better physical interpretation of L'evy flight statistics on a quantum mechanical level. We start with the SFQM Schr"odinger equation characterized by a L'evy flight index \ensuremathα\ensuremath∈(1,2)\ensuremath{\alpha}\ensuremath{\in}(1,2)\ensuremathα\ensuremath∈(1,2), perform a Wigner transform, and draw the limit \ensuremathℏE\ensuremathτ\ensuremath→0\frac{\ensuremath{\hbar}}{E\ensuremath{\tau}}\ensuremath{\rightarrow}0E\ensuremathτ\ensuremathℏ​\ensuremath→0 (i.e., let the observed energy scale EEE go to infinity in comparison to the quantization given by \ensuremathℏ/\ensuremathτ\ensuremath{\hbar}/\ensuremath{\tau}\ensuremathℏ/\ensuremathτ). In order to obtain classical transport equations two possible substitutions for the terms ∣p∣\ensuremathα{|p|}^{\ensuremath{\alpha}}∣p∣\ensuremathα and ∣p\ensuremath′∣\ensuremathα|{p}^{\ensuremath{'}}{|}^{\ensuremath{\alpha}}∣p\ensuremath′∣\ensuremathα which appear in von Neumann's equation are presented. It is demonstrated that they conform to the criteria for a successful Wigner transform. Their benefits and caveats are discussed in detail. We find, that, indeed, SFQM manifests itself in an anomalous kinetic term of the free particle's motion and, assuming an external potential diagonal in momentum space for the sake of simplicity, in corresponding anomalous terms in the resulting drift current. All our results reduce to the classical forms in the limit \ensuremathα=2\ensuremath{\alpha}=2\ensuremathα=2.
A time fractional quantum framework has been introduced into quantum mechanics.A new version of the space-time fractional Schrödinger equation has been launched.The introduced space-time fractional Schrödinger equation has a new scale parameter, which is a fractional generalization of Planck's constant in quantum physics.It has been shown that the presence of a time fractional time derivative in the space-time fractional Schrödinger equation significantly impacts quantum mechanical fundamentals.Time fractional quantum mechanical operators of coordinate, momentum and angular momentum were defined and their commutation relationships were established.The pseudo-Hamilton operator was introduced and its Hermicity has been proven.Two new functions related to the Mittag-Leffler function have been introduced to solve the space-time fractional Schrödinger equation.Energy of a time fractional quantum system has been defined and calculated in terms of the newly introduced functions.It has been found that in the framework of time fractional quantum mechanics there are no stationary states, and the eigenvalues of the pseudo-Hamilton operator are not the energy levels of the time fractional quantum system.A free particle solution to the space-time fractional Schrödinger equation was found.A free particle space-time fractional quantum mechanical kernel has been found and expressed in terms of the H-function.Renormalization properties of a free particle solution and the space-time fractional quantum kernel were established.Some particular cases of time fractional quantum mechanics have been analyzed and discussed.
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In this thesis, the recent semiclassical approach to quantum Brownian motion of Coffey et\nal. [Phys. Chem. Chem. Phys. 9, 3361 (2007)] is studied in detail. This approach is based\non a semiclassical extension of Wigner's method of obtaining corrections to the classical\nequilibrium Maxwell-Boltzmann distribution, yielding a master equation for the reduced\nWigner function in quasi-phase space valid in the weak coupling and high temperature\nlimits. In the non-inertial or high damping limit, this master equation reduces to a quantum\nSmoluchowksi equation (QSE), where the drift coefficient remains as in the classical case,\nwhile the diffusion coefficient is altered to include the quantum parameter and derivatives\nof the potential.
"Fractional quantum mechanics is a recently emerged and rapidly developing field of quantum physics. This is the first monograph on fundamentals and physical applications of fractional quantum mechanics, written by its founder.The fractional Schrödinger equation and the fractional path integral are new fundamental physical concepts introduced and elaborated in the book. The fractional Schrödinger equation is a manifestation of fractional quantum mechanics. The fractional path integral is a new mathematical tool based on integration over Lévy flights. The fractional path integral method enhances the well-known Feynman path integral framework. Related topics covered in the text include time fractional quantum mechanics, fractional statistical mechanics, fractional classical mechanics and the α-stable Lévy random process. The book is well-suited for theorists, pure and applied mathematicians, solid-state physicists, chemists, and others working with the Schrödinger equation, the path integral technique and applications of fractional calculus in various research areas. It is useful to skilled researchers as well as to graduate students looking for new ideas and advanced approaches."--Publisher's website
In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of ℏ can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of ℏ.
A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.