### Abstract

A simple, general method is derived for evaluating the second- and third-order WKB energy integrals by rewriting the integrals having nonintegrable singularities in terms of derivatives, with respect to the energy, of integrals having integrable singularities. As an example, it is shown that the higher-order WKB integrals vanish for the one-dimensional linear harmonic oscillator. A calculation of some eigenvalues using this method is made for potentials of the form V(x)=λx
^{2y} and the results are compared to the "exact" results obtained from a numerical integration of the Schrödinger equation. It is observed that inclusion of the third-order integral improves the accuracy of WKB eigenvalues.

Original language | English (US) |
---|---|

Pages (from-to) | 2942-2945 |

Number of pages | 4 |

Journal | The Journal of Chemical Physics |

Volume | 47 |

Issue number | 8 |

State | Published - 1967 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*The Journal of Chemical Physics*,

*47*(8), 2942-2945.

**Use of the WKB method for obtaining energy eigenvalues.** / Krieger, J. B.; Lewis, M. L.; Rosenzweig, Carl.

Research output: Contribution to journal › Article

*The Journal of Chemical Physics*, vol. 47, no. 8, pp. 2942-2945.

}

TY - JOUR

T1 - Use of the WKB method for obtaining energy eigenvalues

AU - Krieger, J. B.

AU - Lewis, M. L.

AU - Rosenzweig, Carl

PY - 1967

Y1 - 1967

N2 - A simple, general method is derived for evaluating the second- and third-order WKB energy integrals by rewriting the integrals having nonintegrable singularities in terms of derivatives, with respect to the energy, of integrals having integrable singularities. As an example, it is shown that the higher-order WKB integrals vanish for the one-dimensional linear harmonic oscillator. A calculation of some eigenvalues using this method is made for potentials of the form V(x)=λx 2y and the results are compared to the "exact" results obtained from a numerical integration of the Schrödinger equation. It is observed that inclusion of the third-order integral improves the accuracy of WKB eigenvalues.

AB - A simple, general method is derived for evaluating the second- and third-order WKB energy integrals by rewriting the integrals having nonintegrable singularities in terms of derivatives, with respect to the energy, of integrals having integrable singularities. As an example, it is shown that the higher-order WKB integrals vanish for the one-dimensional linear harmonic oscillator. A calculation of some eigenvalues using this method is made for potentials of the form V(x)=λx 2y and the results are compared to the "exact" results obtained from a numerical integration of the Schrödinger equation. It is observed that inclusion of the third-order integral improves the accuracy of WKB eigenvalues.

UR - http://www.scopus.com/inward/record.url?scp=36849102082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36849102082&partnerID=8YFLogxK

M3 - Article

VL - 47

SP - 2942

EP - 2945

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 8

ER -