四阶精度差分法解定态薛定谔方程

Solving time-independent Schr dinger equation by the fourth-order accurate difference method

利用有限差分法数值求解定态薛定谔方程时,文献中常用的是3点中心差分公式,其截断误差为步长的二次方量级,无法满足高精度要求.本文利用多项式插值法,取最近邻和次近邻节点做5点插值,给出导数的四阶精度差分公式,并用于求解几个常见势场中的定态方程.计算结果表明,相对于二阶精度的中心差分,四阶精度差分收敛更快,在相同步长下得到的结果更加精确.

In the finite difference calculations of the time-independent Schr dinger equation,the mostly used difference formula is the central difference formula,which is accompanied with a truncation error on the second-order of step-size.In this paper,the fourth-order accurate difference formulas of the derivatives are derived by the five-point polynomial interpolation,and used to solve time-independent Schr dinger equation in several common potential wells.The numerical results show that,the fourth-order accurate difference formula has better convergence and higher precision than the common central difference formula.