位势问题的Crank-Nicholson方法精确数值解
Precise Numerical Solutions of Potential Problems Using Crank-Nicholson Method
摘要：
本文提出了一种新的数值处理方法，即在Crank-Nicholson方法中引入虚时间演化算子，以求解薛定谔方程。将原有的时间演化技术推广到一个新的算符，该算符提供了一种系统的方法，不仅可以计算基态的本征值，而且可以计算激发态的本征值。这种新的方法系统地产生特征值的精度为十一位数的康奈尔势，覆盖非微扰制度。实现了基于功率计数规则的绝对误差估计技术。这种方法是检查与完全可解的问题，并产生的数值精度下降到10 $^{-11}$。
英文摘要：
A new numerical treatment in the Crank-Nicholson method with the imaginary time evolution operator is presented in order to solve the Schrödinger equation. The original time evolution technique is extended to a new operator that provides a systematic way to calculate not only eigenvalues of ground state but also of excited states. This new method systematically produces eigenvalues with accuracies of eleven digits with the Cornell potential that covers non-perturbative regime. An absolute error estimation technique based on a power counting rule is implemented. This method is examined with exactly solvable problems and produces the numerical accuracy down to 10$^{-11}$.
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