分数阶量子力学下的二维无限深方势阱

数学物理学报 ›› 2022, Vol. 42 ›› Issue (4): 1018-1026.

分数阶量子力学下的二维无限深方势阱

谭云杰^1,²(),韩小惠^1,²(),董建平^1,^2,*()

¹ 南京航空航天大学数学学院南京 211106
² 飞行器数学建模与高性能计算工业和信息化部重点实验室南京航空航天大学南京 211106

收稿日期:2021-03-10 出版日期:2022-08-26 发布日期:2022-08-08
通讯作者: 董建平 E-mail:tanyunjie@nuaa.edu.cn;hanxiaohui@nuaa.edu.cn;dongjianping@nuaa.edu.cn
作者简介:谭云杰, E-mail: tanyunjie@nuaa.edu.cn|韩小惠, E-mail: hanxiaohui@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专项资金(NZ2019008)

Two-Dimensional Infinite Square Well in Fractional Quantum Mechanics

Yunjie Tan^1,²(),Xiaohui Han^1,²(),Jianping Dong^1,^2,*()

¹ College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106
² Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Najing 211106

Received:2021-03-10 Online:2022-08-26 Published:2022-08-08
Contact: Jianping Dong E-mail:tanyunjie@nuaa.edu.cn;hanxiaohui@nuaa.edu.cn;dongjianping@nuaa.edu.cn
Supported by:
the NSFC(11701278);the Fundamental Research Funds for the Central Universities(NZ2019008)

摘要/Abstract

摘要：

分数阶量子力学是标准量子力学的一种推广, 由包含分数阶Riesz导数的空间分数阶Schrödinger方程所描述. 该文考虑了在二维无限深方势阱中运动的自由粒子, 利用Lévy路径积分传播子, 得到了在二维无限深方势阱内运动粒子的波函数和能量本征值. 此外, 运用Lévy路径积分摄动技术, 得到了在$\delta$函数摄动下的二维无限深方势阱区域内运动粒子的能量依赖格林函数.

关键词: 分数阶Schrödinger方程, Lévy路径积分, 传播子, 无限深方势阱

Abstract:

Fractional quantum mechanics is a generalization of standard quantum mechanics, which is described by fractional Schrödinger equation with fractional Riesz derivative operator. In this paper, we consider a free particle moving in a two-dimensional infinite square well, By using Lévy path integral method, the wave function and energy eigenvalue of the two-dimensional infinite square well are obtained. Then the perturbation expansion method is used to study the two-dimensional infinite square well with $\delta$ function, and the corresponding energy-dependent Green's function is obtained.

Key words: Fractional Schrödinger equation, Lévy path integral, Propagator, Infinite square well

中图分类号:

O174.3

谭云杰,韩小惠,董建平. 分数阶量子力学下的二维无限深方势阱[J]. 数学物理学报, 2022, 42(4): 1018-1026.

Yunjie Tan,Xiaohui Han,Jianping Dong. Two-Dimensional Infinite Square Well in Fractional Quantum Mechanics[J]. Acta mathematica scientia,Series A, 2022, 42(4): 1018-1026.

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