Fractional Quantum Mechanics

The following sections are included:

• Contents
• Preface

The following sections are included:

• Path integral
• Fractals and fractional calculus
• Lévy flights
• Schrödinger equation
• Physical applications of fractional quantum mechanics
• Book outline

The term fractal is now used as a scientific concept, as well as a physical object. Fractal is a geometric shape that is self-similar on all scales. In other words, no matter how much you magnify a fractal, it always looks the same (or at least similar). Fractals are generally irregular (not smooth) in shape, and thus are not objects definable by traditional geometry. That means that fractals tend to have significant details, visible at any arbitrary scale. In other words, ‘zooming in’ to a fractal simply shows similar pictures, and it is so-called self-similarity. For example, a normal ‘Euclidean’ shape, such as a circle, looks flatter and flatter as it is magnified. At infinite magnification it is impossible to tell the difference between a circle and a straight line. Fractals are not like this. The conventional idea of curvature, which represents the reciprocal of the radius of an approximating circle, cannot be usefully applied because it scales away. Instead, in a fractal, increasing the magnification reveals details that you simply couldn’t see before. The secondary characteristics of fractals, while intuitively appealing, are remarkably hard to condense into a mathematically precise definition. Strictly, a fractal should have a fractional, that is, non-integer dimension. Fractals can be deterministic or random. The examples of deterministic fractals include the Mandelbrot set [27], Lyapunov fractal, Cantor set, Sierpinski carpet and triangle, Peano curve, and the Koch snowflake [28]. Examples of random fractals are the Brownian motion and Lévy flights and their generalizations [27]…

The following sections are included:

• Fundamentals
• Fractional Schrödinger equation in coordinate representation
  • Quantum Riesz fractional derivative
• Fractional Schrödinger equation in momentum representation
• Hermiticity of the fractional Hamiltonian operator
• The parity conservation law in fractional quantum mechanics
• Velocity operator
• Fractional current density

The special case when Hamiltonian H_α does not depend explicitly on time is of great importance for physical applications. It is easy to see that in this case there exists a special solution to the 3D fractional Schrödinger equation (3.8) in the form

Heisenberg’s Uncertainty Principle [101], which is a fundamental aspect of quantum mechanics, has many different mathematical formulations. The uncertainty relation is a mathematical formulation of Heisenberg’s Uncertainty Principle. The first uncertainty relation in the framework of standard quantum mechanics was developed by Kennard [102] in 1927. Since then many other mathematical approaches have been developed to formulate Heisenberg’s Uncertainty Principle…

The following sections are included:

• Quantum kernel
• Phase space representation
• Coordinate representation
  • 3D coordinate representation
• Feynman’s path integral
  • 3D generalization of Feynman’s path integral
• Fractional Schrödinger equation from the path integral over Lévy flights
  • 3D fractional Schrödinger equation from the path integral over Lévy flights

The following sections are included:

• Fundamental properties
  • Scaling
• Fox H-function representation for a free particle kernel
  • A free particle kernel: 3D case
  • A free particle kernel: D-dimensional generalization

The following sections are included:

• Laplace transform
• The energy-time transformation
• A free particle Green function
• Fractional kernel in momentum representation

The following sections are included:

• Quarkonium and quantum fractional oscillator
  • Fractional oscillator in momentum representation
• Spectrum of 1D quantum fractional oscillator in semiclassical approximation
  • Coordinate representation
  • Momentum representation
• Dimensionless fractional Schrödinger equation for fractional oscillator
• Symmetries of quantum fractional oscillator

The following sections are included:

• A free particle
  • Scaling properties of the 1D fractional Schrödinger equation
  • Exact 1D solution
  • Exact 3D solution
• Infinite potential well
  • Consistency of the solution for infinite potential well
• Bound state in δ-potential well
• Linear potential field
• Quantum kernel for a free particle in the box
• Fractional Bohr atom

The nonlinear Schrödinger equation is an attractive and fast developing area of studies. Not pretending to give a complete list of publications on the topic, let us mention some of them, such as [123]-[129].

Our intent is to show how nonlinear fractional Schrödinger equation and fractional Ginzburg–Landau equation emerge from quantum dynamics with long-range interaction.

The following sections are included:

• Introductory remarks
  • Shortcomings of time fractional quantum mechanics
  • Benefits of time fractional quantum mechanics
• Time fractional Schrödinger equation
  • Naber approach
• Space-time fractional Schrödinger equation
  • Wang and Xu, and Dong and Xu approach
  • Laskin approach
• Pseudo-Hamilton operator in time fractional quantum mechanics
• 3D generalization of space-time fractional Schrödinger equation
  • Hermiticity of pseudo-Hamilton operator
  • The parity conservation law
  • Space-time fractional Schrödinger equation in momentum representation
• Solution to space-time fractional Schrödinger equation
• Energy in the framework of time fractional quantum mechanics

The following sections are included:

• A free particle wave function
• Infinite potential well problem in time fractional quantum mechanics
• A free particle space-time fractional quantum mechanical kernel
  • Renormalization properties of the space-time fractional quantum mechanical kernel
  • Wright L-function in time fractional quantum mechanics
  • Fox H-function representation for a free particle space-time fractional quantum mechanical kernel
• Special cases of time fractional quantum mechanics
  • The Schrödinger equation
  • Fractional Schrödinger equation
  • Time fractional Schrödinger equation

The following sections are included:

• Density matrix
  • Phase space representation
  • Coordinate representation
  • Motion equation for the density matrix
  • Fundamental properties of a free particle density matrix
  • Scaling of density matrix: a free particle
  • Density matrix for a particle in a one-dimensional infinite well
• Partition function
• Density matrix in 3D space
  • Coordinate representation
  • A free particle partition function in 3D
  • Feynman’s density matrix
• Fractional thermodynamics
  • Ideal gas
  • Ideal Fermi and Bose gases

Fractional classical mechanics was introduced by Laskin [134] as a classical counterpart of fractional quantum mechanics. Here we present the Lagrange, Hamilton and Hamilton–Jacobi frameworks for fractional classical mechanics. Scaling analysis of fractional classical motion equations has been implemented based on the mechanical similarity. We discover and discuss fractional Kepler’s third law which is a generalization of the well-known Kepler’s third law…

Following [159] we develop an alternative approach to study fractional classical mechanical oscillator described by the Hamilton function given by Eq. (15.41). To simplify the notations we rewrite Eq. (15.41) in the unified form to cover 1D and 2D cases,

The idea behind the approach is to transform an integrable Hamiltonian system with two degrees of freedom on the plane into a dynamic system that is defined on the sphere and inherits the integrals of motion of the original system…

The following sections are included:

• Afterword
• Appendix A: Fox H-function
  • Definition of the Fox H-function
  • Fundamental properties of the H-function
    • Cosine transform of the H-function
    • Some functions expressed in terms of H-function
    • Lévy α-stable distribution in terms of H-function
• Appendix B: Fractional Calculus
  • Brief introductory remarks
    • Definition of the Riemann–Liouville fractional derivative
    • Definition of Caputo fractional derivative
    • Definition of quantum Riesz fractional derivative
• Appendix C: Calculation of the Integral
• Appendix D: Polylogarithm
  • Polylogarithm as a power series
  • Properties of function $\mathcal{V}\left(k\right)$
• Appendix E: Fractional Generalization of Trigonometric Functions cos z and sin z
• Bibliography
• Index