Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Multidimensional tunneling in terms of complex classical mechanics: Wave functions, energy splittings, and decay rates in nonintegrable systems

Shoji Takada

Published Mar 8, 1996• 28 citations
Publisher

Abstract

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.

DOI
10.1063/1.471029

Related papers

Semiclassical theory of multidimensional tunneling and the hopping method

Gennady Mil’nikov, António J. C. Varandas

The paper presents a new semiclassical theory of multidimensional tunneling and its application to the decay problem. A simple semiclassical expression for the decay rate constant is derived in terms of complex valued family of classical trajectories in the decay valley while the algebraic form of the family in the tunneling region is found by means of classical canonical perturbation theory. This provides a tool to analytically continue the classical trajectories into the decay valley, with a numerical illustration of such a continuation being done for a model 2D decay rate problem. The calculated results for the rate constant are found to be in good agreement with the exact ones for the high levels where the suggested perturbative treatment is applicable. It is also shown that the formulated theory can be directly compared with a previously proposed hopping method which gives a way to examine the accuracy of the latter without exact quantum calculations.

The Journal of Chemical Physics•Nov 8, 1999•13 citations

Wentzel–Kramers–Brillouin theory of multidimensional tunneling: General theory for energy splitting

Shoji Takada, Hiroki Nakamura

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 Journal of Chemical Physics•Jan 1, 1994•98 citations

Semiclassical construction of Green's functions for multidimensional systems with barrier tunneling

Kenneth G. Kay

We develop a treatment that divides phase space into regions bounded by surfaces and forms the energy-dependent semiclassical Green's function from short classical trajectory segments confined to the individual regions. We use this to generalize Bogomolny's quantization method to systems with barrier tunneling. Combined with an appropriate treatment of tunneling amplitudes, this yields a unified semiclassical framework that uses ordinary, real trajectories in classically allowed regions and imaginary-time trajectories in forbidden regions to obtain energies, decay rates, and wave functions for arbitrary states of multidimensional systems undergoing tunneling. Because the trajectory segments in the allowed region are short, components in the Green's function expression can be numerically evaluated using semiclassical initial value methods, yielding a promising approach for practical calculations.

Physical Review A•Oct 27, 2017•5 citations

Instantons re-examined: Dynamical tunneling and resonant tunneling

Jérémy Le Deunff, Amaury Mouchet

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.

arXiv:0911.4093•Physical Review E•Apr 8, 2010•26 citations

Instantons revisited: dynamical tunnelling and resonant tunnelling

Jérémy Le Deunff, Amaury Mouchet

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.

•Apr 8, 2010•0 citations

WKB Theory of Tunneling between Tori

Shoji Takada, Hiroki Nakamura

General WKB theory of multidimensional tunneling between tori is formulated and illuminating physical picture on the effects of multidimensionality is provided. On the basis of Wilkinson's work, (1) Maslov's semiclassical wavefunction in classically accessible region is connected to the wavefunction in classically inaccessible region and (2) the latter is propagated into deep tunneling region. It is found that there exist two distinct regions of tunneling : a region where action is pure imaginary (I-region) and another region where action is complex (C-region). In I-region tunneling path can be defined by a certain classical trajectory on the inverted potential, while in C-region no tunneling path can be defined and the Huygens type wave propagation should be carried out. Moreover, wavefunction of a vibrationally excited state has nodal lines in C-region but not in I-region.

Progress of Theoretical Physics Supplement•Feb 1, 1994•3 citations
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.
The paper presents a new semiclassical theory of multidimensional tunneling and its application to the decay problem. A simple semiclassical expression for the decay rate constant is derived in terms of complex valued family of classical trajectories in the decay valley while the algebraic form of the family in the tunneling region is found by means of classical canonical perturbation theory. This provides a tool to analytically continue the classical trajectories into the decay valley, with a numerical illustration of such a continuation being done for a model 2D decay rate problem. The calculated results for the rate constant are found to be in good agreement with the exact ones for the high levels where the suggested perturbative treatment is applicable. It is also shown that the formulated theory can be directly compared with a previously proposed hopping method which gives a way to examine the accuracy of the latter without exact quantum calculations.
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.
We develop a treatment that divides phase space into regions bounded by surfaces and forms the energy-dependent semiclassical Green's function from short classical trajectory segments confined to the individual regions. We use this to generalize Bogomolny's quantization method to systems with barrier tunneling. Combined with an appropriate treatment of tunneling amplitudes, this yields a unified semiclassical framework that uses ordinary, real trajectories in classically allowed regions and imaginary-time trajectories in forbidden regions to obtain energies, decay rates, and wave functions for arbitrary states of multidimensional systems undergoing tunneling. Because the trajectory segments in the allowed region are short, components in the Green's function expression can be numerically evaluated using semiclassical initial value methods, yielding a promising approach for practical calculations.
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.
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.
General WKB theory of multidimensional tunneling between tori is formulated and illuminating physical picture on the effects of multidimensionality is provided. On the basis of Wilkinson's work, (1) Maslov's semiclassical wavefunction in classically accessible region is connected to the wavefunction in classically inaccessible region and (2) the latter is propagated into deep tunneling region. It is found that there exist two distinct regions of tunneling : a region where action is pure imaginary (I-region) and another region where action is complex (C-region). In I-region tunneling path can be defined by a certain classical trajectory on the inverted potential, while in C-region no tunneling path can be defined and the Huygens type wave propagation should be carried out. Moreover, wavefunction of a vibrationally excited state has nodal lines in C-region but not in I-region.