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

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 th...

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Bibliographic Details
Published in:Journal of computational physics 2008-02, Vol.227 (5), p.2970-2976
Main Authors: Kang, Daekyoung, Won, E.
Format: Article
Language:English
Subjects:
Computational techniques
Crank–Nicolson method
Exact sciences and technology
Finite differences
Imaginary time
Mathematical methods in physics
Physics
Precise numerical calculation
Schrödinger equation
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Summary: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 .
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2007.11.028