Journal ArticleDOI
Some physical applications of fractional Schrödinger equation
Xiaoyi Guo,Mingyu Xu +1 more
TLDR
In this article, the fractional Schrodinger equation was solved for a free particle and for an infinite square potential well, and the energy levels and the normalized wave functions of a particle in a potential well were obtained.Abstract:
The fractional Schrodinger equation is solved for a free particle and for an infinite square potential well. The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained. In the barrier penetration problem, the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined. In the quantum scattering problem, according to the fractional Schrodinger equation, the Green’s function of the Lippmann-Schwinger integral equation is given.read more
Citations
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Journal ArticleDOI
Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation
TL;DR: In this article, an accurate spectral collocation method for solving one-and two-dimensional variable-order fractional nonlinear cable equations is presented. But the method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobic operational matrix for variable-orders derivatives, described in the sense of Caputo.
Journal ArticleDOI
An energy conservative difference scheme for the nonlinear fractional Schrödinger equations
Pengde Wang,Chengming Huang +1 more
TL;DR: An energy conservative Crank–Nicolson difference scheme for nonlinear Riesz space-fractional Schrodinger equations is studied and the existence of the difference solution is proved based on Brouwer fixed point theorem.
Journal ArticleDOI
Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative
TL;DR: The Crank-Nicolson (CN) difference scheme for the coupled nonlinear Schrodinger equations with the Riesz space fractional derivative is studied and the existence of this difference solution is proved by the Brouwer fixed point theorem.
Journal ArticleDOI
A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations
TL;DR: An iterative algorithm is proposed, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners.
Journal ArticleDOI
Space-time fractional Schrödinger equation with time-independent potentials
Jianping Dong,Mingyu Xu +1 more
TL;DR: In this article, a space-time fractional Schrodinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schröder equation was developed.
References
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Book
The Fractal Geometry of Nature
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Book
Quantum Mechanics and Path Integrals
TL;DR: Au sommaire as discussed by the authors developed the concepts of quantum mechanics with special examples and developed the perturbation method in quantum mechanics and the variational method for probability problems in quantum physics.
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Scattering theory of waves and particles
TL;DR: In this paper, the authors present a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect, and the general approach to multiparticle reaction theory.
Journal ArticleDOI
Fractional Schrödinger equation.
TL;DR: The Hermiticity of the fractional Hamilton operator and the parity conservation law for fractional quantum mechanics are established and the energy spectra of a hydrogenlike atom and of a fractional oscillator in the semiclassical approximation are found.