显式与隐式方法求解含时薛定谔方程及误差分析  

EXPLICIT AND IMPLICIT NUMERICAL METHODS FOR TIME-DEPENDENT SCHRÖDINGER EQUATION AND ERROR ANALYSIS

作　　者：郑纾寒 潘超钰 陈保义 ZHENG Shuhan;PAN Chaoyu;CHEN Baoyi(School of Science,Tianjin University,Tianjin 100875)

机构地区：[1]天津大学理学院,天津100875

出　　处：《物理与工程》2024年第1期57-61,共5页Physics and Engineering

基　　金：国家自然科学基金项目(12175165)。

摘　　要：含时薛定谔方程是量子力学最重要的方程之一,它可以给出不同相互作用势下体系波函数的演化。相互作用势的复杂形式使得薛定谔方程一般没有解析解。如何较准确地数值求解含时薛定谔方程,对许多物理问题有着重要意义。本文采用显式与隐式的方法求解薛定谔方程。从结果可以发现,隐式的方法得到的波函数精度远高于显式方法,且误差具有收敛性。为了进一步探索隐式格式的可行性,本文还采用有限温度下的屏蔽势,利用隐式方法具体求解粲夸克偶素的波函数演化。As one of the most significant equations in Quantum Mechanics,time-dependent Schrodinger equation can describe the evolution of wave function in different interaction potential.The complicated forms of interaction potential make it generally impossible to find an analytical solution to the Schrödinger equation.Therefore,how to accurately numerically solve the time-dependent Schrödinger equation is of great significance for many physical problems.In this paper,emphasis is placed on how to solve Schrödinger equation with both explicit and implicit numerical methods,where the conclusion can be drawn that the wave function obtained by implicit method is more accurate than that of the explicit method based on the outcome of calculation.What's more,the deviation of implicit method can be proven to be convergent.In order to further explore the feasibility of the implicit scheme,this paper also adopts the screened potential at finite temperature and uses the implicit method to specifically solve the wave function evolution of charmonium.

关 键 词：含时薛定谔方程 偏微分方程数值求解 显隐式格式稳定性分析 

分 类 号：O413.1[理学—理论物理]