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CRC Press eBooks

Multidimensional Theory of Tunneling Splitting

Hiroki Nakamura, Gennady Mil'nikov

Published Jul 18, 2013• 1 citation
Publisher

Abstract

In this section, generalizations of the instanton theory and the modified WKB theory described in Chapter 2 to multidimensional space are presented [43]. Those who are not interested in the generalization of the instanton approach and the proof of its equivalence to the modified WKB theory can skip Section 6.1.1 and Section 6.1.2. In Section 6.1.3 a general WKB formulation for a general Hamiltonian in curved space is provided and its final expression of tunneling splitting can be directly applied to any real systems.

DOI
10.1201/b14673-10

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In this section, generalizations of the instanton theory and the modified WKB theory described in Chapter 2 to multidimensional space are presented [43]. Those who are not interested in the generalization of the instanton approach and the proof of its equivalence to the modified WKB theory can skip Section 6.1.1 and Section 6.1.2. In Section 6.1.3 a general WKB formulation for a general Hamiltonian in curved space is provided and its final expression of tunneling splitting can be directly applied to any real systems.
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.
It is rather obvious that one-dimensional theory is not good enough to comprehend quantum mechanical tunneling phenomena. For instance, the tunneling splitting estimated using the one-dimensional potential along the intrinsic reaction coordinate gives a few orders of magnitude smaller value [56]. There is a more crucial example that demonstrates the conceptual deficiency of one-dimensional treatment and shows interesting features peculiar to multidimensional tunneling phenomena. This is explained in Chapter 4. In this sense, multidimensional treatment is definitely needed. Herewe describe the two-dimensionalWKB theory developed by Takada and Nakamura [30].
Starting from trace formulae for the tunnelling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instanton-like approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunnelling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunnelling or resonant tunnelling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunnelling.
In this paper we present a method for obtaining the WKB approximation for non-separable multidimensional potentials. At its leading order one has to solve the classical Hamilton-Jacobi equation with zero energy for which a very efficient method is proposed. The essence of this method lies in the recognition that in the vicinity to the potential minimum the solution of Schrödinger equation has to approach that of coupled harmonic oscillators. Quantum corrections to the semiclassical result are then obtained very easily and to an arbitrarily high order. The method is applied to the calculation of the tunneling splitting and tunneling lifetime. We show that classical turning points are part of the dynamical problem; they could not be determined simply from looking at the form of the potential, but are obtained from the solution of the pertinent equations of motion.
Starting from trace formulas for the tunneling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instantonlike approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunneling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunneling or resonant tunneling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunneling.