Fractional Calculus And Its Applications To Fractional Quantum Mechanics

The paper focuses on fractional calculus and its applications to fractional quantum mechanics. It is composed of four chapters. The first chapter contains a brief introduction to fractional calculus and some elementary knowledge. The second and third chapters deal with the space fractional quantum system described by a space fractional Schr(?)dinger equation. In the last chapter, some properties of the space-time fractional quantum system described by a space-time fractional Schr(?)dinger equation are presented.In chapter 1, the history and the development of the fractional calculus and its applications is introduced. We also introduce some kinds of fractional operators, such as Riemann-Liouville, Caputo and Riesz fractional operators. Additionally, some special functions, which are the elementary solutions of many fractional differential equations, are shown. We present the Mittag-Leffler function and generlized Mittag-Leffler function in two, three and four parameters. Moreover, we give the definition and some properties of the Fox H-function, which includes nearly all the special functions occurring in applied mathematics and statistics as its special cases, such as Mittag-Leffler type function, generalized hyperge-omatric function, generalized Bessel function, Meijer's G-function and so on.In chapter 2, we study the one-dimensional space fractional Schr(?)dinger equation with linear potential, delta-function potential and Coulomb potential. With the help of Fourier transformation, the space fractional Schr(?)dinger equation under momentum representation is obtained. By use of Mellin transform and its inverse transform, we obtain the energy levels and wave functions expressed in H-function for a particle in linear potential field. The wave function expressed also by H-function and the unique energy level of the bound state for the particle of even parity state in delta-function potential well, which is proved to have no action on the particle of odd parity state, is also obtained. The integral form of the wave functions for a particle in Coulomb potential field is shown and the corresponding energy levels which have been discussed in Laskin's paper [Phys. Rev. E 66, 056108 (2002)] are proved to satisfy an equality of infinite limit of H-function. All of these results contain the ones of the standard quantum mechanics as their special cases.Furthermore, by means of dimensional analysis, we give a specific mathematical expression for the undetermined constant D_αin fractional quantum mechanics. Then we plot two figures to show the effect, of the changing ofα(1 <α≤2) which also denotes the fractal dimension of the Levy-like quantum mechanical path, on the wave functions of a particle in linear potential field andδ-potential well, respectively. We find that when a takes a smaller value, the wave function for the linear potential oscillates more rapidly with higher amplitude and shorter period, and the wave function for theδ-potential well decreases more rapidly.In chapter 3, we find that the continuity or discontinuity condition of a fractional derivative, defined by V_α=(-h²â–³)^α/2-1(?), of the wave functions should be considered to solve the fractional Schrodinger equation in fractional quantum mechanics, just as the continuity or discontinuity of the lst-order derivative of the wave functions is often used to solve the standard Schr(?)dinger equation in standard quantum mechanics. If the potential function is finite in any fixed region, V_αφ(x)(φ(x) denotes the wave function) is continuous everywhere, but if the potential function is infinite at some points, V_αφ(x) is discontinuous at these points as long as the wave function is not zero in the vicinity of the points. According to theses conclusions, the space fractional Schrodinger equations with a finite square potential, periodic potential, and delta-function potential are studied.In standard quantum mechanics, there is only one kind of even or odd parity state for a finite square potential well. However, we obtain two kinds of wave functions for each of the parity states for fractional quantum mechanics. The energy equations for these states are also obtained and solved graphically.Block's theorem for fractional quantum mechanics with periodic potential fields is obtained and the energy for the periodic potential exhibits a "band structure" just as with standard quantum mechanics. When studying the energy band structure, we find that, the 1st-order derivative of the wave function is not continuous when there is more than one linear-independent real solution to the fractional Schrodinger equation for periodic potential.The jump (discontinuity) condition for the fractional derivative of the wave function of theδ-potential is given. With the help of the jump condition, we study some kinds ofδ-potential fields. For theδ-potential well, an alternate expression of the wave function, in terms of elementary functions, is derived. It is more useful and efficient than that in terms of H function given in chapter 2. Then, the problem of a particle penetrating through a delta-function potential barrier and the fractional probability current density of the particle are also discussed. We prove that the fractional probability current density is continuous everywhere even though V_αφ(x) is discontinuous at the origin. At the end of this chapter, we study the Dirac comb, and the energy band structure for the Dirac comb is shown in figure 3.3, from which we come to a conclusion that the admitted energy regions get more narrow for largerα.In chapter 4, we develop a space-time fractional Schr(?)dinger equation containing Ca-puto fractional derivative and the quantum Riesz fractional operator from the space fractional Schr(?)dinger equation given by Laskin. The new space-time fractional Schr(?)dinger equation is given bywhereBy use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schr(?)dinger equation with time-independent poten-tials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained.With the help of the properties of fractional operators, we study the time evolution laws of the space-time fractional quantum system. The time evolution formulas of mechanical quantities are obtained and from them we find that there is no conservative mechanical quantities of motion for the space-time fractional quantum system. A Mittag-Leffler type of time evolution operator of wave functions and a Heisenberg equaion containing fractional operators are also given by us. All of these results are the generalization of those in the standard quantum mechanics.On studying the time evolution of the space-time fractional quantum system in the time-independent potential fields, we find that the time evolution properties of the quantum system not only depend on the order of time fractional derivative, but also are affected by the sign of the eigenvalue of the space equation. When the eigenvalue of the space equation is positive, the total probability and the energy levels reach some limiting values as time evolutes and the limiting value of the total probability may be less or greater than one, which means the potential may absorb or release particles but the absorbing or releasing behavior becomes weaker enough as time progresses so that the quantum system approaches some steady states with fixed nonzero particle probability and energy levels. Moreover, the limiting value of the total probability can never be zero when 0 <β< 1 (βdenotes the order of the time fractional derivative, as mentioned before) but may be zero when 1 <β< 2 in some special cases (see the conclusions given in section§4.4, on page 60 of this paper). Therefore, when the eigenvalue of the space equation is positive, only for 1 <β< 2, the particles in the space-time fractional quantum system may be absorbed completely by the potential. When the eigenvalue of the space equation is negative, the time limits of the total probability and the energy levels are zeros when 0 <β< 1 or infinities when 1 <β< 2, which means the potential absorbs particles completely when 0 <β< 1 but releases particles all the time when 1 <β< 2 and the quantum system will never come to a steady state in the latter case.From the conclusions given before, we know that the basic characteristics of all of the time fractional quantum systems in the time-independent potential fields is that the probability is not conservative no matter whether the space term is fractional or not. In other words, the introduction of the time fractional derivative to the Schrodinger equation causes non-conservation of probability.