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

Journal of Physics A Mathematical and General

Asymptotic Green functions: a generalized semiclassical approach for scattering by multiple barrier potentials

M. G. E. da Luz, Bin Kang Cheng, Marcus W. Beims

Published Jun 7, 2001• 17 citations
Publisher

Abstract

We show how quantum mechanical barrier reflection and transmission coefficients ℛ and 𝒯 can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V ( x ) = ∑ j V ( j ) ( x ), where each V ( j ) goes to zero for x →±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual ℛ ( j ) and 𝒯 ( j ) . The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.

DOI
10.1088/0305-4470/34/24/303

Related papers

Multiple scattering approach to one-dimensional potential problems

Arlen Anderson

By calculating the reflection and transmission coefficients for a square potential barrier in terms of a series of scatterings at step potential discontinuities, a multiple scattering approach to potential problems is presented. This approach is applied to WKB potential problems by interpreting the WKB connection formulas in terms of reflection and transmission coefficients at each turning point. An algorithm for the systematic construction of the energy Green’s function is given. The significance of complex coordinate turning points is demonstrated. The equivalence of this approach and that of the Gaussian approximation of the path integral is mentioned briefly.

American Journal of Physics•Mar 1, 1989•30 citations

A generalized semiclassical expression for the eigenvalues of multiple well potentials

Fabiano M. Andrade, Bin Kang Cheng, Marcus W. Beims +1 more

From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.

Journal of Physics A Mathematical and General•Dec 12, 2002•16 citations

Semiclassical path representation of the Green function in one-dimensional multiple-well potentials

Walter T. Strunz

Using semiclassical approximations and graph-theoretical concepts, a path representation of the energy-dependent Green function in a one-dimensional multiple-well potential is derived. Applying uniform semiclassical methods, this representation is valid for energies close to the barrier maxima, where tunnelling and above-barrier reflection are of importance. Semiclassical quantization is discussed and a closed path representation of the density of states is derived. Applications to wavepacket dynamics and to the quantization of disordered systems are presented.

Journal of Physics A Mathematical and General•Jul 7, 1992•9 citations

Semiclassically weak reflections above analytic and non-analytic potential barriers

Michael Berry

The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If V(x) is analytic, all terms have the same semiclassical order (exp(-h(cross) -1 )); the multiple integrals are evaluated exactly and the series summed. If V(x) has a discontinuous Nth derivative, the term m=1 dominates semiclassically and gives r approximately h(cross) N . If V(x) has all derivatives continuous but possesses an essential singularity on the real axis, the term m=1 again dominates semiclassically, and for V approximately exp(- mod x mod -n ) gives r approximately exp(-h(cross) -n (n+1)/) with an oscillatory factor corresponding to transmission resonances. The formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de Broglie wavelength is less than 1% of the barrier width and mod r mod 2 approximately 10 -1000 .

Journal of Physics A Mathematical and General•Dec 1, 1982•97 citations

Nonlocal approach to scattering in a one-dimensional problem

Ph. Grossel, J. M. Vigoureux, Fadi Baida

An alternative approach to the one-dimensional scattering problem is presented. We show that the reflection and the transmission coefficients of any number of quantum wells or barriers can directly be obtained by using two mathematical methods. The first method arises from a generalization of the so-called elementary symmetric functions used in the mathematical theory of polynomials; the second one consists of a complex generalization of Einstein's addition law for parallel velocities. These laws for the reflection and transmission amplitude coefficients permit a rapid evaluation of the transmittivity of any kind of potential barrier. Some examples are considered. Results are compared with the ones obtained with the WKB method. We also present a numerical study to test the stability of the method.

Physical Review A•Nov 1, 1994•27 citations

Semiclassical Real-Time Tunneling by Multiple Spawning of Classical Trajectories

Frank Großmann

We systematically extend semiclassical real-time propagation methods based on Gaussian wave packets by using the composition property of the time-dependent quantum-mechanical Green function. For the first time, in an application to the problem of barrier tunneling, a semiclassical time-domain calculation of the transmission probability exhibits good agreement with exact quantum mechanical values at energies below the barrier top. Only two insertions of unity are needed for these results in a benchmark model of reactive scattering.

Physical Review Letters•Jul 31, 2000•42 citations
We show how quantum mechanical barrier reflection and transmission coefficients ℛ and 𝒯 can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V ( x ) = ∑ j V ( j ) ( x ), where each V ( j ) goes to zero for x →±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual ℛ ( j ) and 𝒯 ( j ) . The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.
By calculating the reflection and transmission coefficients for a square potential barrier in terms of a series of scatterings at step potential discontinuities, a multiple scattering approach to potential problems is presented. This approach is applied to WKB potential problems by interpreting the WKB connection formulas in terms of reflection and transmission coefficients at each turning point. An algorithm for the systematic construction of the energy Green’s function is given. The significance of complex coordinate turning points is demonstrated. The equivalence of this approach and that of the Gaussian approximation of the path integral is mentioned briefly.
From the poles of a generalized semiclassical Green's function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.
Using semiclassical approximations and graph-theoretical concepts, a path representation of the energy-dependent Green function in a one-dimensional multiple-well potential is derived. Applying uniform semiclassical methods, this representation is valid for energies close to the barrier maxima, where tunnelling and above-barrier reflection are of importance. Semiclassical quantization is discussed and a closed path representation of the density of states is derived. Applications to wavepacket dynamics and to the quantization of disordered systems are presented.
The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If V(x) is analytic, all terms have the same semiclassical order (exp(-h(cross) -1 )); the multiple integrals are evaluated exactly and the series summed. If V(x) has a discontinuous Nth derivative, the term m=1 dominates semiclassically and gives r approximately h(cross) N . If V(x) has all derivatives continuous but possesses an essential singularity on the real axis, the term m=1 again dominates semiclassically, and for V approximately exp(- mod x mod -n ) gives r approximately exp(-h(cross) -n (n+1)/) with an oscillatory factor corresponding to transmission resonances. The formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de Broglie wavelength is less than 1% of the barrier width and mod r mod 2 approximately 10 -1000 .
An alternative approach to the one-dimensional scattering problem is presented. We show that the reflection and the transmission coefficients of any number of quantum wells or barriers can directly be obtained by using two mathematical methods. The first method arises from a generalization of the so-called elementary symmetric functions used in the mathematical theory of polynomials; the second one consists of a complex generalization of Einstein's addition law for parallel velocities. These laws for the reflection and transmission amplitude coefficients permit a rapid evaluation of the transmittivity of any kind of potential barrier. Some examples are considered. Results are compared with the ones obtained with the WKB method. We also present a numerical study to test the stability of the method.
We systematically extend semiclassical real-time propagation methods based on Gaussian wave packets by using the composition property of the time-dependent quantum-mechanical Green function. For the first time, in an application to the problem of barrier tunneling, a semiclassical time-domain calculation of the transmission probability exhibits good agreement with exact quantum mechanical values at energies below the barrier top. Only two insertions of unity are needed for these results in a benchmark model of reactive scattering.