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CRC Press eBooks

Multidimensional Effects: Peculiar Phenomena

Hiroki Nakamura, Gennady Mil'nikov

Published Jul 18, 2013• 0 citations
Publisher

Abstract

As was pointed out in Section 3.1, there are some interesting features in multidimensional tunneling compared with the one-dimensional case [30,31,46]. These are the following: (1) There exist two types of tunneling, pure tunneling and mixed tunneling. (2) No tunneling path can be defined in the case of mixed tunneling. (3) Vibrational excitation can suppress tunneling because of mixed tunneling. In this section, these peculiar features are numerically demonstrated and analyzed [31] by taking tunneling splitting in symmetric double well potential as an example. Three typical two-dimensional model potentials are used.

DOI
10.1201/b14673-8

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As was pointed out in Section 3.1, there are some interesting features in multidimensional tunneling compared with the one-dimensional case [30,31,46]. These are the following: (1) There exist two types of tunneling, pure tunneling and mixed tunneling. (2) No tunneling path can be defined in the case of mixed tunneling. (3) Vibrational excitation can suppress tunneling because of mixed tunneling. In this section, these peculiar features are numerically demonstrated and analyzed [31] by taking tunneling splitting in symmetric double well potential as an example. Three typical two-dimensional model potentials are used.
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].
Tunneling is a supreme quantum effect. Every introductory text [1] on quantum mechanics gives the paradigm example of a particle tunneling through a one-dimensional potential barrier despite having a total energy less than the barrier height. Indeed, the reader typically works through a number of exercises, all involving one-dimensional potential barriers of one form or another modeling several key physical phenomena ranging from atom transfer reactions to the decay of α-particles [2]. However, one seldom encounters coupled multidimenisonal tunneling in such texts since an analytical solution of the Schro¨dinger equation in such cases is not possible. Interestingly, the richness and complexity of the tunneling phenomenon manifest themselves in full glory in the case of multidimensional systems [3]. Thus, for instance, the usual one-dimensional expectation of increasing tunneling splittings as one approaches the barrier top from below is not necessarily true as soon as one couples another bound degree of freedom to the tunneling coordinate. In the context of molecular reaction dynamics, multidimensional tunneling can result in strong mode-specificity and fluctuations in the reaction rates [4]. In fact, a proper description of tunneling of electrons and hydrogen atoms is absolutely essential [5,6] even in molecular systems as large as enzymes and proteins. Although one usually assumes tunneling effects to be significant in molecules involving light atom transfers, it is worth pointing out that neglecting the tunneling of even a heavy atom like carbon is the difference between a reaction occurring or not occurring. In particular, one can underestimate rates by nearly hundred orders of magnitude [7]. Interestingly, and perhaps paradoxically, several penetrating insights into the nature and mechanism of multidimensional barrier tunneling have been obtained from a phase-space perspective [8,9]. The contributions by Creagh, Shudo and Ikeda, and Takahashi in the present volume provide a detailed account of the latest advances in the phasespace-based understanding of multidimensional barrier tunneling.
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 tunneling effect is the most popular phenomenon of quantum physics and is present in modern physical theories. Still, the most important features of this effect are already present in toy models - low dimensional quantum mechanics with high potential barriers. Even these simple systems cannot be solved analytically and approximations have to be made. Tunneling is closely related to splitting of the ground state energy which is degenerate in the classical limit. In this thesis we use two methods to estimate the energy splitting in one dimensional quantum mechanics. In semi-classical approximation the splitting is calculated in the weak coupling limit. The other approach, Tamm-Dancoff method gives very precise numerical results for finite coupling. Comparing these two approaches we are able to confirm results of the semi-classical approximation and determine range of its applicability. Several potentials are considered: anharmonic double well, cosine potential and cosine potential in periodic space with finite number of minima. We also address a non-generic problem of anharmonic triple well potential.
One of the appealing features of tunneling as a theoretical problem is that it poses interesting questions regarding the complexified dynamics of the classical limit. Often, these questions are of a different nature to the ones that arise in a purely dynamical systems context and can therefore push the dynamics side of the problem in novel directions, without necessarily being technically hard. In this article we explore a number of scenarios where these underlying problems of complex dynamics can be well enough understood to allow explicit analytical approximations to be given for tunneling rates. The treatment here is not intended to be comprehensive and short shrift is given to very interesting and important tunneling regimes, such as those of resonance-and chaos-assisted tunneling [3,14,16,17,40,48,78], for example. Instead, by concentrating on relatively tractable problems where answers can be given in terms of simple geometrical characterizations of the complex dynamics, we aim to achieve an understanding of problems which are important in their own right but which may also later serve as building blocks in the solution of more difficult problems.