一维含时薛定谔方程的MATLAB有限差分矩阵分解算法

Matrix Decomposition Algorithm for One-dimensional Time-dependent Schr(o|¨)dinger Equation with Finite Difference and MATLAB

介绍了一种用于求解一维含时薛定谔方程的MATLAB矩阵分解算法。首先用等间距步长将距离和时间分为一系列的离散节点。其次,用向后差分近似表示时间导数,用中心差分近似表示空间导数,由此可导出一维含时薛定谔方程的古典隐差分格式。在不同的初始条件或初始/边界条件下,它们可以转化成一个用矩阵方程表示的节点线性方程组。在每一个时间步长,利用MATLAB提供的矩阵左除命令即可求出各个未知节点的函数近似值。重复该过程,便可获得任意时间步长下的波函数值。该方法概念简单,使用方便,无需在编程上花费较多精力即可求解一维含时薛定谔方程。

A MATLAB matrix decomposition algorithm for solving the one-dimensional time-dependent Schrodinger equation is presented.Firstly,the distance and time are divided into a series of discrete nodes with evenly spaced increments.Secondly,the time derivative is approximated by backward difference and the space derivative is approximated by central difference,so a classic implicit format difference scheme that is absolutely stable for the one-dimensional time-dependent Schr（o｜¨）dinger equation can be derived.Subsequently,they can be converted into the nodal linear equations which can be expressed in a matrix equation under different initial or initial/boundary conditions.For each time increment,the approximation of each unknown nodal function can be solved with the matrix left division command in MATLAB.The procedure may be repeated to obtain the wave function values for any desired number of time increments.It is simple in concept,convenient in operation and can be used to solve the one-dimensional time-dependent Schrodinger equation without more efforts in programming.