Precise numerical solutions of potential problems using the Crank-Nicolson method

Abstract

We present a numerically precise treatment of the Crank-Nicolson method with an imaginary time evolution operator in order to solve the Schrodinger equation. The time evolution technique is applied to the inverse-iteration method that provides a systematic way to calculate not only eigenvalues of the ground-state but also of the excited-states. This method systematically produces eigenvalues with the accuracy of eleven digits when the Cornell potential is used. An absolute error estimation technique is implemented based on a power counting rule. This method is examined on exactly solvable problems and produces the numerical accuracy down to 10(-11). (c) 2007 Elsevier Inc. All rights reserved.