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arXiv:hep-th/9808137 · Physical Review A

Double-well potential: The WKB approximation with phase loss and anharmonicity effect

Chang Soo Park, Myung Geun Jeong, Sahng-Kyoon Yoo, D.K. Park

Published Nov 1, 1998• 28 citations
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Abstract

We derive a general WKB energy splitting formula in a double-well potential by incorporating both phase loss and the anharmonicity effect in the usual WKB approximation. A bare application of the phase loss approach to the usual WKB method gives better results only for large separation between two potential minima. In the range of substantial tunneling, however, the phase loss approach with the anharmonicity effect considered leads to a great improvement in the accuracy of the WKB approximation.

DOI
10.1103/physreva.58.3443

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We derive a general WKB energy splitting formula in a double-well potential by incorporating both phase loss and the anharmonicity effect in the usual WKB approximation. A bare application of the phase loss approach to the usual WKB method gives better results only for large separation between two potential minima. In the range of substantial tunneling, however, the phase loss approach with the anharmonicity effect considered leads to a great improvement in the accuracy of the WKB approximation.
The accuracy of the WKB approximation when predicting the energy splitting of bound states in a double well potential is the main subject of this paper. The splitting of almost degenerate energy levels below the top of the barrier results from the tunneling and is thus supposed to be exponentially small. By using the standard WKB quantization we deduce an analytical formula for the energy splitting, which is the usual Landau formula with additional quantum corrections. We also examine the accuracy of our and Landau formula numerically for the case of the symmetric double well quartic potential.
An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1
We study the WKB method away from the short-wave limit. Incorporating the correct phase loss due to reflection at a classical turning point, in place of the usual choice $\ensuremath{\pi}/2$, can greatly improve the accuracy of the WKB wave function in the classically allowed region. For a repulsive 1/x2{1/x}^{2}1/x2 potential this leads to far more accurate WKB wave functions that the usual Langer modification of the potential.