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Journal of Physics A Mathematical and General

A generalized semiclassical expression for the eigenvalues of multiple well potentials

Fabiano M. Andrade, Bin Kang Cheng, Marcus W. Beims, M. G. E. da Luz

Published Dec 12, 2002• 16 citations
Publisher

Abstract

From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.

DOI
10.1088/0305-4470/36/1/315

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From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.
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 show how quantum mechanical barrier reflection and transmission coefficients ℛ and 𝒯 can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V ( x ) = ∑ j V ( j ) ( x ), where each V ( j ) goes to zero for x →±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual ℛ ( j ) and 𝒯 ( j ) . The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.
Using semiclassical approximations and graph-theoretical concepts, a path representation of the energy-dependent Green function in a one-dimensional multiple-well potential is derived. Applying uniform semiclassical methods, this representation is valid for energies close to the barrier maxima, where tunnelling and above-barrier reflection are of importance. Semiclassical quantization is discussed and a closed path representation of the density of states is derived. Applications to wavepacket dynamics and to the quantization of disordered systems are presented.
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.
A semiclassical theory of multidimensional tunneling is formulated to calculate the tunneling wave function, energy splitting in a double well and decay rate constant from a metastable state. First, the tunneling wave function is calculated by analytic continuation of a quantized torus prepared in analytic forms using either the Chapman–Garrett–Miller method or the Birkhoff–Gustavson normal form method. For a weakly nonintegrable system, tunnelings are confirmed to be classified into two qualitatively different domains; pure tunneling in the I region and mixed tunneling in the C region. Semiclassical wave functions agree with quantum mechanical ones within a few percent both in classically allowed and tunneling regions. Breakdown of this simple picture is exemplified for cases of relatively strong couplings. Second, expressions of the tunneling energy splitting in a double well potential and the decay rate from a metastable state are derived. The wave function near a well formulated above is connected with the semiclassical Green’s function in the deep tunneling region. The latter is expressed by complex trajectories which start from the complex quantized torus. A preliminary numerical comparison with the quantum mechanical value is also done for the energy splitting.