分数阶薛定谔方程的积分形式及其在量子散射中的应用.docVIP

分数阶薛定谔方程的积分形式及其在量子散 射中的应用 董建平 南京航空航天大学 理学院 数学系，南京　 210016 摘要：本文给出了三维分数阶薛定谔方程的积分形式，并应用于分数阶量子力学框架下散射问 题的研究。利用玻恩近似方法，本文导出了分数阶散射问题对应的波函数和散射波幅的解析近 似解。此外，文中还给出了可进行任意阶修正的玻恩级数形式的解。最后，本文分析了玻恩近 似方法在分数阶量子体系中的可行性，导出了其适用的条件。 关键词：应用数学，分数阶薛定谔方程，Riesz 分数阶导数，量子散射，玻恩近似. 中图分类号： O29 Integral Form of the Fractional Schr?dinger Equation and Its Application to the Quantum Scattering Problems DONG Jian-ping Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016 Abstract: The integral form of the 3D space fractional Schr?dinger equation is derived in this paper. Using this integral equation, we study the scattering problems in the fractional quantum mechanics and obtain the analytical approximate solutions of the wave function and the scattering amplitude using the Born approximation method. The Born series, with corrections of every order, is also given. In the end, we discuss the validity of the Born approximation and show the validity condition. Key words: Applied mathematics，Fractional Schr?dinger equation, Riesz fractional derivative, Quantum scattering, Born approximation. 基金项目： This work was supported by the National Natural Science Foundation of China (Grant No., the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20113218120030), and the Fundamental Research Funds for the Central Universities (Grant No. NS2012119). 作者简介： Dong Jianping（1982-），male，associate professor，major research direction：fractional calculus and its applications. -1- 0 Introduction Nowadays, the fractional calculus [1–3] becomes a popular tool for scientists. It has been used successfully in various ?elds of science and engineering, and the fractional di?erential equations become very popular for describing anomalous transport, di?usion-reaction processes, super-slow relaxation, classical mechanics, etc. [4–9] (and the references therein). In recent years, the fractional calculus enters the world of qu