Precise Numerical Solutions of Potential Problems Using Crank-Nicholson Method

Abstract

A new numerical treatment in the Crank-Nicholson method with the imaginary time evolution operator is presented in order to solve the Schr\"{o}dinger 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$^{-11}$.