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arXiv:1703.05758

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

Seyyed M. H. Halataei, Anthony J. Leggett

Published Mar 15, 2017• 6 citations
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Abstract

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.

DOI
10.48550/arxiv.1703.05758

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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.
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
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 double-well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of 'classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double-well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel–Kramers–Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but also for the excited states.
Despite the fact that quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, I derive a succinct analytical formula based on the Wentzel-Kramers-Brillouin semiclassical approach. Two different examples of asymmetric potentials are discussed: the cases when the two localized levels are degenerate and when they are not degenerate. For the first case, I also discuss a time-dependent problem showing the quantum Zeno effect.