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Fox's H-Functions and the Time Fractional Schr\"odinger Equation

Selçuk Ş. Bayın

Published Mar 16, 2011• 0 citations
OpenAlex

Abstract

After introducing the formalism of the general space and time fractional Schrodinger equation, we concentrate on the time fractional Schrodinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time fractional Schrodinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is less than one and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem. PACS numbers: this http URL, 02.50.Ey, this http URL, 03.65.Db

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After introducing the formalism of the general space and time fractional Schrodinger equation, we concentrate on the time fractional Schrodinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time fractional Schrodinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is less than one and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem. PACS numbers: this http URL, 02.50.Ey, this http URL, 03.65.Db
After introducing the formalism of the general space and time fractional Schrödinger equation, we concentrate on the time fractional Schrödinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time-fractional Schrödinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is ⩽1 and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem.
We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.
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.
Recently, it was pointed out that the solutions found in the literature for the space fractional Schrödinger equation in a piecewise manner are wrong, except the case with the delta potential. We re-analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schrödinger equation in general coordinates in terms of Fox's H-functions.
In this study, the effect of time fractionalization on the development of quantum systems is taken under consideration by making use of fractional calculus. In this context, a Mittag–Leffler function is introduced as an important mathematical tool in the generalization of the evolution operator. In order to investigate the time fractional evolution of the quantum (nano) systems, time fractional forms of motion are obtained for a Schrödinger equation and a Heisenberg equation. As an application of the concomitant formalism, the wave functions, energy eigenvalues, and probability densities of the potential well and harmonic oscillator are time fractionally obtained via the fractional derivative order α, which is a measure of the fractality of time. In the case α=1, where time becomes homogenous and continuous, traditional physical conclusions are recovered. Since energy and time are conjugate to each other, the fractional derivative order α is relevant to time. It is understood that the fractionalization of time gives rise to energy fluctuations of the quantum (nano) systems.
The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order.The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [𝑀𝐿 2 /𝑇 2 ].The action 𝑆 is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced.The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order.The present work corrects a number of errors in Naber's work.The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained.The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity.A generalization to the case of a particle moving in a potential is also given.