有限差分法求解一维非线性谐振子问题

Solving the One-Dimensional Nonlinear Harmonic Oscillator Problem by Finite Difference Method

针对许多量子体系很难得到薛定谔方程解析解这个问题,本文提出采用有限差分法求解薛定谔方程,从而将连续的量子力学本征值问题转化为离散的矩阵运算问题.首先,以一维线性谐振子为例,采用有限差分法求解了该体系的本征能量以及本征函数;然后,与一维线性谐振子的解析解进行对比,验证了有限差分法求解薛定谔方程的可行性与准确性;最后,又采用有限差分法求解了一维非线性谐振子的本征能量以及本征函数,并与微扰法得到的近似解进行了比较.

Aiming at the problem that it is difficult to obtain the analytical solution of the Schr dinger equation for many quantum systems,this paper proposes the finite difference method to solve the Schr dinger equation,thereby transforming the continuous eigenvalue problem of quantum mechanics into a discrete matrix operation problem.Firstly,taking the one-dimensional linear harmonic oscillator as an example,the eigenenergy and the eigenfunction of the system are solved by the finite difference method.Then compared with the analytical solution of the one-dimensional linear harmonic oscillator,the feasibility and accuracy of the finite difference method to solve the Schr dinger equation is verified.Finally,the finite difference method is used to solve the eigenenergy and eigenfunction of the one-dimensional nonlinear harmonic oscillator,and they are compared with the approximate solutions obtained by the perturbation method.