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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Multidimensional tunneling and complex momentum

Peter Bowcock, Ruth Gregory

Published Sep 15, 1991• 21 citations
Publisher

Abstract

We examine the problem of modeling tunneling phenomena in more than one dimension. We find that existing techniques are inadequate in a wide class of situations because of their inability to deal with concurrent classical motion. We show how to generalize these methods to allow for complex momenta, and demonstrate the improved techniques with a selection of illustrative examples. We comment on possible applications.

DOI
10.1103/physrevd.44.1774

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In the case of tunneling in chemical reactions [see figure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the first-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions [59]. Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as briefly explained there, but cannot be simply extended to multidimensions (see, for instance, [15]). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory [206], in which the quantized boundary conditions are satisfied at both (initial and final) ends. Unfortunately, this method turned out not to be practically useful, because finding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difficulty of detecting caustics in multidimensional space. An efficient way of doing that has been devised along classical trajectory [47]. This is not a trivial task at all, since it is not possible to find caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-fly method. In this subsection, this method is explained.

CRC Press eBooks•Jul 18, 2013•2 citations
We examine the problem of modeling tunneling phenomena in more than one dimension. We find that existing techniques are inadequate in a wide class of situations because of their inability to deal with concurrent classical motion. We show how to generalize these methods to allow for complex momenta, and demonstrate the improved techniques with a selection of illustrative examples. We comment on possible applications.
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.
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.
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.
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.
In the case of tunneling in chemical reactions [see figure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the first-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions [59]. Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as briefly explained there, but cannot be simply extended to multidimensions (see, for instance, [15]). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory [206], in which the quantized boundary conditions are satisfied at both (initial and final) ends. Unfortunately, this method turned out not to be practically useful, because finding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difficulty of detecting caustics in multidimensional space. An efficient way of doing that has been devised along classical trajectory [47]. This is not a trivial task at all, since it is not possible to find caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-fly method. In this subsection, this method is explained.