We present a numerically precise treatment of the Crank-Nicolson method with an imaginary time evolution operator in order to solve the Schrödinger 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.
Bibliographical noteFunding Information:
We thank Jungil Lee for his suggestion on this topic and Q-Han Park and Ki-Hwan Kim for useful discussion on the numerical treatment. E.W. is indebted to Tai Hyun Yoon for his critical comments on this manuscript. D.K.’s research was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD), (KRF-2006-612-C00003). E.W.’s research was supported by Grant No. R01-2005-000-10089-0 from the Basic Research Program of the Korea Science & Engineering Foundation.
- Crank-Nicolson method
- Finite differences
- Imaginary time
- Precise numerical calculation
- Schrödinger equation
ASJC Scopus subject areas
- Numerical Analysis
- Modelling and Simulation
- Physics and Astronomy (miscellaneous)
- General Physics and Astronomy
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics