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

Daekyoung Kang; E. Won

doi:10.1016/j.jcp.2007.11.028

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

Daekyoung Kang, E. Won

Department of Physics

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

Abstract

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}.

Original language	English
Pages (from-to)	2970-2976
Number of pages	7
Journal	Journal of Computational Physics
Volume	227
Issue number	5
DOIs	https://doi.org/10.1016/j.jcp.2007.11.028
Publication status	Published - 2008 Feb 20

Keywords

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)
Physics and Astronomy(all)
Computer Science Applications
Computational Mathematics
Applied Mathematics

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10.1016/j.jcp.2007.11.028

Cite this

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title = "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 Schr{\"o}dinger 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.",

keywords = "Crank-Nicolson method, Finite differences, Imaginary time, Precise numerical calculation, Schr{\"o}dinger equation",

author = "Daekyoung Kang and E. Won",

note = "Funding 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.{\textquoteright}s research was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD), (KRF-2006-612-C00003). E.W.{\textquoteright}s research was supported by Grant No. R01-2005-000-10089-0 from the Basic Research Program of the Korea Science & Engineering Foundation.",

year = "2008",

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doi = "10.1016/j.jcp.2007.11.028",

language = "English",

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TY - JOUR

T1 - Precise numerical solutions of potential problems using the Crank-Nicolson method

AU - Kang, Daekyoung

AU - Won, E.

N1 - Funding 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.

PY - 2008/2/20

Y1 - 2008/2/20

N2 - 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.

AB - 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.

KW - Crank-Nicolson method

KW - Finite differences

KW - Imaginary time

KW - Precise numerical calculation

KW - Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=38549105519&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2007.11.028

DO - 10.1016/j.jcp.2007.11.028

M3 - Article

AN - SCOPUS:38549105519

SN - 0021-9991

VL - 227

SP - 2970

EP - 2976

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 5

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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