Multidimensional WKB Approximation for Tunneling Along Curved Escape Paths |
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Authors: | J Zamastil L Skála |
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Institution: | (1) Department of Chemical Physics and Optics, Charles University, Faculty of Mathematics and Physics, Ke Karlovu 3, 121 16 Prague 2, Prague, Czech Republic;(2) Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada |
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Abstract: | Asymptotics of the perturbation series for the ground state energy of the coupled anharmonic oscillators for the positive
coupling constant is related to the lifetime of the quasistationary states for the negative coupling constant. The latter
is determined by means of the multidimensional WKB approximation for tunneling along curved escape paths. General method for
obtaining such approximation is described. The cartesian coordinates (x,y) are choosen in such a way that the x-axis has the
direction of the probability flux at large distances from the well. The WKB wave function is then obtained by the simultaneous
expansion of the wave function in the coordinate y and the parameter γ determining the curvature of the escape path. It is
argued, both physically and mathematically, that these two expansions are mutually consistent. Several simplifications in
the integrations of equations are pointed out. It is shown that to calculate outgoing probability flux it is not necessary
to deal with inadequacy of the WKB approximation at the classical turning point. The WKB formulas for the large-order behavior
of the perturbation series are compared with numerical results and an excellent agreement between the two is found. |
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Keywords: | multidimensional wkb approximation large-order behavior of the perturbation series |
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