arXiv:quant-ph/9609008

Effect of Anharmonicity on the WKB Energy Splitting in a Double Well Potential

Park, Chang Soo, Soo-Young Lee, Jae-Rok Kahng, Sahng-Kyoon Yoo, D.K. Park, C. H. Lee, Eui-Soon Yim

Published Sep 9, 1996• 0 citations

Abstract

We investigate the effect of anharmonicity on the WKB approximation in a double well potential. By incorporating the anharmonic perturbation into the WKB energy splitting formula we show that the WKB approximation can be greatly improved in the region over which the tunneling is appreciable. We also observe that the usual WKB results can be obtained from our formalism as a limiting case in which the two potential minima are far apart.

DOI: 10.48550/arxiv.quant-ph/9609008

Related papers

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

Chang Soo Park, Myung Geun Jeong, Sahng-Kyoon Yoo +1 more

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.

arXiv:hep-th/9808137Physical Review ANov 1, 199828 citations

Tunneling and energy splitting in an asymmetric double-well potential

Dae-Yup Song

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

Aug 4, 201322 citations

High Order WKB Prediction of the Energy Splitting in the Symmetric Double Well Potential

Marko Robnik, Luca Salasnich, Marko Vrani\v car

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.

arXiv:nlin/0003050Progress of Theoretical Physics SupplementJan 1, 20001 citation

WKB corrections to the energy splitting in double well potentials

Marko Robnik, Luca Salasnich, Vranicar, Marko

By using the WKB quantization we deduce an analytical formula for the energy splitting in a double--well potential which is the usual Landau formula with additional quantum corrections. Then we analyze the accuracy of our formula for the double square well potential, the inverted harmonic oscillator and the quartic potential.

arXiv:nlin/0003062Mar 29, 20002 citations

Tunnel Splitting in Asymmetric Double Well Potentials: An Improved WKB Calculation

Seyyed M. H. Halataei, Anthony J. Leggett

We present an improved Wentzel-Kramers-Brillouin (WKB) calculation of tunnel splitting in one dimensional asymmetric double well potentials. We show that the tunnel splitting in general can have linear dependence on the bias energy, beside the well-known quadratic dependence. We demonstrate that the linear correction is greater than previously thought.

arXiv:1703.05758Mar 15, 20176 citations

Wentzel–Kramers–Brillouin theory of multidimensional tunneling: General theory for energy splitting

Shoji Takada, Hiroki Nakamura

A general Wentzel–Kramers–Brillouin (WKB) theory of multidimensional tunneling is formulated and an illuminating physical picture of the effects of multidimensionality is provided. Two basic problems are solved: (i) Maslov’s semiclassical wave function in the classically accessible region is connected to the wave function in the classically inaccessible region and (ii) the latter is propagated into the deep tunneling region. It is found that there exist two distinct types of tunneling: pure tunneling and mixed tunneling. The former is the usual one in which the tunneling path can be defined by a certain classical trajectory on the inverted potential and its associated action is pure imaginary. In the latter case, no tunneling path can be defined and the Huygens-type wave propagation should be carried out. In this case, tunneling is always accompanied by classical motion in the transversal direction and the associated action is complex. A general procedure is presented for the evaluation of energy splitting ΔE in the double well. Moreover, under the locally separable linear approximation, a simple and convenient formula for ΔE is derived and is confirmed to work well by comparison with the exact numerical calculations.

The Journal of Chemical PhysicsJan 1, 199498 citations