Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Some physical applications of fractional Schrödinger equation

Xiaoyi Guo, Mingyu Xu

Published Aug 1, 2006• 268 citations
Publisher

Abstract

The fractional Schrödinger 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 Schrödinger equation, the Green’s function of the Lippmann-Schwinger integral equation is given.

DOI
10.1063/1.2235026

Related papers

Wave Mechanics and Its Simple Applications

Ajay Sharma

The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.

CRC Press eBooks•Jun 18, 2021•0 citations

Green's functions in quantum mechanics courses

William J. Herrera, Herbert Vinck-Posada, Shirley Gómez Páez

The use of Green's functions is valuable when solving problems in electrodynamics, solid-state physics, and many-body physics. However, its role in quantum mechanics is often limited to the context of scattering by a central force. This work shows how Green's functions can be used in other examples in quantum mechanics courses. In particular, we introduce time-independent Green's functions and the Dyson equation to solve problems with an external potential. We calculate the reflection and transmission coefficients of scattering by a Dirac delta barrier and the energy levels and local density of states of the infinite square well potential.

arXiv:2107.14104•American Journal of Physics•Sep 22, 2022•6 citations

Application of Green’s-Function techniques to quantum barrier penetration problems

Rahul K Singh, Durgesh Dixena, Dwas Lal +1 more

We obtain a mathematical and algebraic expression of the Green-function method in treating quantum tunnelling in one-dimensional barrier systems. The retarded Green's function of the free-particle resolvent is then obtained consistently to obtain scattering amplitudes using the Lippmann-Schwinger equation. The technique is used to apply to the d-function barrier, the finite square barrier, and the double-barrier quantum-well structure in order to describe the transmission, penetration of the barrier, and resonance. With the Green, the pole structure in the case of the double-barrier directly traces the quasi-bound and the Breit-Wigner-type resonances.

International Journal of Physics and Mathematics•Jul 1, 2025•0 citations

Uniform operator: Aligning fractional time quantum mechanics with basic physical principles

Xueling Duan, Cheng Ma, Hongyin Huang +1 more

In this paper, we construct a fractional time corresponding operator to derive the uniform fractional time Schrödinger equation, which implies the relationship between the fractionaloperatorandtheenergy. Weverifythreebasicprinciplesoftheuniformfractional time Schrödinger equation: (a) Unitarity of the time evolution operator; (b) Existence of stationary energy spectra; (c) Probability conservation law. We apply the uniform equation to study three typical particles: free particle, an infinite potential well and a rectangular barrier. This uniform equation brings some typical physical phenomena, including (1) non-symmetry basic wave functions; (2) continuous energy spectrum for a free particle and an infinite potential well; (3) The barrier penetration in the rectangular barrier. These phenomena were not found in previous researches and the standard quantum mechanics, which will attract further applications in various fields of science.

International Journal of Modern Physics A•Mar 1, 2025•3 citations

Fractional Quantum Problem Based on Caputo-Fabrizio Derivative

Bouzenna Fatma El-Genbazi, Meftah Mohammed T

Background and Objective: In the context of the novel definition of the fractional derivative, we solve the Fractional Schrödinger equation based on the non-singular Caputo-Fabrizio derivative definition.The quantum particle is submitted to a step potential.We focus our study to find the reflection and transmission coefficients.Materials and Methods: To reach this goal, we first work out the fractional differential equation defined in terms of the Caputo-Fabrizio derivative.Results: Hence, according to the obtained wave functions of fractional Schrödinger equations, the reflection and transmission coefficient are calculated.Conclusion: Interestingly, our findings collapse to the standard results when the fractional order of derivative become integer numbers.

Journal of Applied Sciences•Jan 15, 2022•0 citations

Infinite wall in the fractional quantum mechanics

Jianping Dong, Ying Lu

The space-fractional Schrödinger equation for a local potential is difficult to solve because the fractional Riesz operator in it is nonlocal. In fractional quantum mechanics, the infinite wall is a basic problem and has not been solved yet. In this paper, we consider a free particle in an infinite wall region. Making use of the Lévy path integral method, we derive the Lévy path integral amplitude of the particle and take advantage of it to get the solution of the fractional Schrödinger equation for an infinite wall. Then, the infinite wall potential with a delta-function perturbation is also studied. According to Fox’s H-function, we get an equation of bound state energies of a free particle moving in an infinite wall region perturbed by the δ-function for two situations, E < 0 and E > 0. We give a transcendental equation that determines energy levels when E < 0 and prove that there is no bound states when E > 0. The asymptotic behaviors of the center of the delta function approaching to the origin and the infinite are also shown. The results of this paper include special cases in standard quantum mechanics.

Journal of Mathematical Physics•Mar 1, 2021•2 citations
The fractional Schrödinger 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 Schrödinger equation, the Green’s function of the Lippmann-Schwinger integral equation is given.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
The use of Green's functions is valuable when solving problems in electrodynamics, solid-state physics, and many-body physics. However, its role in quantum mechanics is often limited to the context of scattering by a central force. This work shows how Green's functions can be used in other examples in quantum mechanics courses. In particular, we introduce time-independent Green's functions and the Dyson equation to solve problems with an external potential. We calculate the reflection and transmission coefficients of scattering by a Dirac delta barrier and the energy levels and local density of states of the infinite square well potential.
We obtain a mathematical and algebraic expression of the Green-function method in treating quantum tunnelling in one-dimensional barrier systems. The retarded Green's function of the free-particle resolvent is then obtained consistently to obtain scattering amplitudes using the Lippmann-Schwinger equation. The technique is used to apply to the d-function barrier, the finite square barrier, and the double-barrier quantum-well structure in order to describe the transmission, penetration of the barrier, and resonance. With the Green, the pole structure in the case of the double-barrier directly traces the quasi-bound and the Breit-Wigner-type resonances.
In this paper, we construct a fractional time corresponding operator to derive the uniform fractional time Schrödinger equation, which implies the relationship between the fractionaloperatorandtheenergy. Weverifythreebasicprinciplesoftheuniformfractional time Schrödinger equation: (a) Unitarity of the time evolution operator; (b) Existence of stationary energy spectra; (c) Probability conservation law. We apply the uniform equation to study three typical particles: free particle, an infinite potential well and a rectangular barrier. This uniform equation brings some typical physical phenomena, including (1) non-symmetry basic wave functions; (2) continuous energy spectrum for a free particle and an infinite potential well; (3) The barrier penetration in the rectangular barrier. These phenomena were not found in previous researches and the standard quantum mechanics, which will attract further applications in various fields of science.
Background and Objective: In the context of the novel definition of the fractional derivative, we solve the Fractional Schrödinger equation based on the non-singular Caputo-Fabrizio derivative definition.The quantum particle is submitted to a step potential.We focus our study to find the reflection and transmission coefficients.Materials and Methods: To reach this goal, we first work out the fractional differential equation defined in terms of the Caputo-Fabrizio derivative.Results: Hence, according to the obtained wave functions of fractional Schrödinger equations, the reflection and transmission coefficient are calculated.Conclusion: Interestingly, our findings collapse to the standard results when the fractional order of derivative become integer numbers.
The space-fractional Schrödinger equation for a local potential is difficult to solve because the fractional Riesz operator in it is nonlocal. In fractional quantum mechanics, the infinite wall is a basic problem and has not been solved yet. In this paper, we consider a free particle in an infinite wall region. Making use of the Lévy path integral method, we derive the Lévy path integral amplitude of the particle and take advantage of it to get the solution of the fractional Schrödinger equation for an infinite wall. Then, the infinite wall potential with a delta-function perturbation is also studied. According to Fox’s H-function, we get an equation of bound state energies of a free particle moving in an infinite wall region perturbed by the δ-function for two situations, E < 0 and E > 0. We give a transcendental equation that determines energy levels when E < 0 and prove that there is no bound states when E > 0. The asymptotic behaviors of the center of the delta function approaching to the origin and the infinite are also shown. The results of this paper include special cases in standard quantum mechanics.