Wentzel-Kramers-Brillouin method in multidimensional tunneling

Huang, ZhaoHui; Feuchtwang, T. E.; Cutler, P.H.; Kazes, E.

doi:10.1103/physreva.41.32

Cited by 69 publications

(41 citation statements)

References 35 publications

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“…The semiclassical solution for multidimensional tunneling becomes quite involved for non-separable potentials [28]. One must then resort to a numerical solution of the Hamilton-Jacobi equation, although there is still some simplification for the rare occasion when all classical trajectories cross the turning surfaces at right angle [29].…”

Section: Resultsmentioning

confidence: 99%

“…This is, however, not required as long as we are only interested in the duration of the tunneling process. The classically forbidden motion (28) in z-direction is sometimes called an imaginary-time solution, and may be thought of as a classical particle moving with positive energy -E, in the inverted potential F,.…”

Section: Let Us Put (24) and (25) In (23) By Applying The Crucial Rementioning

confidence: 99%

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Tunneling dynamics in stationary field emission

Gottlieb

Kleber

1992

Annalen der Physik

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Abstract. Stationary three-dimensional field emission out of an ultrathin quantum well is analyzed in terms of the semiclassical Hamilton-Jacobi theory. We compare the spatial quantum distribution of the tunneling electrons with the corresponding results obtained from real-time tunneling trajectories. The classical-limit trajectory method reproduces the exact quantum distribution for weak external fields. The Biittiker-Landauer expression for the barrier crossing velocity is confirmed.

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Section: Resultsmentioning

confidence: 99%

Section: Let Us Put (24) and (25) In (23) By Applying The Crucial Rementioning

confidence: 99%

Tunneling dynamics in stationary field emission

Gottlieb

Kleber

1992

Annalen der Physik

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“…In addition, by ] W @) * using eqn. (13), the expression in square brackets in eqn. (11) is reduced to Lm@/Lm which ends the proof.…”

Section: Tunnelling Probability In the Wkb Approximationmentioning

confidence: 99%

“…Mathematically, such a limitation is due to the fact that the complex-valued solution of the HJ equation is described in terms of not one but two coupled sets of EulerÈ Lagrange equations which are not equivalent to a single set of ordinary di †erential equations. 13 In collisional problems, complex classical trajectories have been used to calculate Smatrix elements for classically forbidden processes.21,22 Thus, semiclassical S-matrix theory is formally free from the abovementioned drawback, although the extension of the path integral formalism to complex phase space and its relation with complex classical mechanics needs a more rigorous mathe-matical foundation. Besides, it requires the proper analytical behavior of the potential energy surface in complex coordinate space, which cannot be warranted for most existing models.…”

Section: Introductionmentioning

confidence: 99%

Classical canonical transformation theory as a tool to describe multidimensional tunnelling in reactive scattering. Hopping method revisited and collinear H+H2 exchange reaction near the classical threshold

Mil’nikov¹,

Varandas²

1999

Phys. Chem. Chem. Phys.

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Classical canonical perturbation theory is applied in the vicinity of the saddle point for a chemical reaction. This is done by applying successive canonical transformations in the scope of the GustavsonÈBirkho † approach. It is shown that the calculated approximate classical integrals of motion can be used to describe classically forbidden tunnelling processes. They are also organically embedded into a hopping method to incorporate tunnelling e †ects into classical trajectory simulations of chemical reactions. The applicability of the proposed scheme is demonstrated for the collinear exchange reaction using the double many-body H ] H 2 expansion potential energy surface.

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“…The action S(r) is complex under the barrier for real r. This complexity has also an important counterpart in the instanton formulation of the problem of tunneling decay in a magnetic field, see below. We note that a complex action arises also in other cases, like barrier penetration for oblique incidence 28 and scattering by a complex potential (as in the case of an absorbing medium) 29 . The method discussed below can be applied to these problems as well.…”

Section: U( R)mentioning

confidence: 99%

Tunneling decay in a magnetic field

Sharpee

Dykman

Platzman³

2002

Phys. Rev. A

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We provide a semiclassical theory of tunneling decay in a magnetic field and a three-dimensional potential of a general form. Because of broken time-reversal symmetry, the standard WKB technique has to be modified. The decay rate is found from the analysis of the set of the particle Hamiltonian trajectories in complex phase space and time. In a magnetic field, the tunneling particle comes out from the barrier with a finite velocity and behind the boundary of the classically allowed region. The exit location is obtained by matching the decaying and outgoing WKB waves at a caustic in complex configuration space. Different branches of the WKB wave function match on the switching surface in real space, where the slope of the wave function sharply changes. The theory is not limited to tunneling from potential wells which are parabolic near the minimum. For parabolic wells, we provide a bounce-type formulation in a magnetic field. The theory is applied to specific models which are relevant to tunneling from correlated two-dimensional electron systems in a magnetic field parallel to the electron layer.

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Wentzel-Kramers-Brillouin method in multidimensional tunneling

Cited by 69 publications

References 35 publications

Tunneling dynamics in stationary field emission

Tunneling dynamics in stationary field emission

Classical canonical transformation theory as a tool to describe multidimensional tunnelling in reactive scattering. Hopping method revisited and collinear H+H2 exchange reaction near the classical threshold

Tunneling decay in a magnetic field

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