时空分数阶量子力学下的δ势阱

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1634-1642.

时空分数阶量子力学下的δ势阱

陆莹,谭云杰,董建平*()

^{南京航空航天大学数学系南京 211106}

收稿日期:2020-08-07 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 董建平 E-mail:dongjianping@nuaa.edu.cn
基金资助:
国家自然科学基金(11701278);中央高校基本科研业务费资助项目(NZ2019008)

δ-Potential in Space-Time Fractional Quantum Mechanics

Ying Lu,Yunjie Tan,Jianping Dong*()

^{Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106}

Received:2020-08-07 Online:2021-12-26 Published:2021-12-02
Contact: Jianping Dong E-mail:dongjianping@nuaa.edu.cn
Supported by:
the NSFC(11701278);the Fundamental Research Funds for the Central Universities(NZ2019008)

摘要/Abstract

摘要：

时空分数阶量子力学由含有Caputo导数和Riesz导数的时空分数阶薛定谔方程所描述，是量子力学的推广，可刻画更为广泛的量子现象.该文研究了时空分数阶量子体系下单δ势阱以及双δ势阱中粒子所满足的一维时空分数阶薛定谔方程，求解出了粒子的波函数和能级.此外，利用积分变换建立了δ势阱中粒子的时空分数阶量子力学路径积分核，并导出了其Fox's H函数形式，构建了时空分数阶薛定谔方程和路径积分之间的联系，为从路径积分角度研究时空分数阶量子力学提供了更多的可能性.

关键词: 时空分数阶薛定谔方程, 量子力学核, Fox's H函数

Abstract:

Space-time fractional quantum mechanics, described by Schrödinger equation with Caputo derivative and Riesz derivative, is a generalization of quantum mechanics and can depict more extensive quantum phenomena. This paper studies the one-dimensional space-time fractional Schrödinger equation for a particle in the single and double δ-potential well, and gives the wave functions and energy levels of the particle. In addition, the space-time fractional quantum mechanical path integrals kernels of a particle in the δ-potential well are established by using integral transformation, and the corresponding Fox's H-function forms are derived, and the relation between space-time fractional Schrodinger equation and path integrals is constructed. It provides more possibilities to study space-time fractional quantum mechanics from the perspective of path integrals.

Key words: Space-time fractional Schrödinger equation, Quantum mechanical kernel, Fox's H-function

中图分类号:

陆莹,谭云杰,董建平. 时空分数阶量子力学下的δ势阱[J]. 数学物理学报, 2021, 41(6): 1634-1642.

Ying Lu,Yunjie Tan,Jianping Dong. δ-Potential in Space-Time Fractional Quantum Mechanics[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1634-1642.

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