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

arXiv:hep-th/9609177 · Journal of Physics A Mathematical and General

Scattering from the potential barrier from the path integration over

H. Ahmedov, I H Duru

Published Jan 7, 1997• 3 citations
arXivHTMLPDF

Abstract

Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and non-compact bases, and new addition theorems are derived which are required in path integral applications involving a positively signed potential. The Green function for the potential barrier is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup. The transition and the reflection coefficients are given. Results for the moving barrier are also presented.

DOI
10.1088/0305-4470/30/1/012

Related papers

SCATTERING FROM THE POTENTIAL BARRIER V = COSH-2OMEGA X FROM THE PATH INTEGRATION OVER SO(1, 2)

H. Ahmedov, I. H. Duru

Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and noncompact basis and the new addition theorems are derived which are required in path integral applications involving the positively signed potential. The Green function for the potential barrier $V=cosh^{-2}\\omega x$ is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup.The transition and the reflection coefficients are given.Results for the moving barrier $V=cosh^{-2}\\omega (x-g_0t)$ are also presented.

Journal of Physics A Mathematical and Theoretical•Sep 23, 1996•0 citations

Path integration over compact and noncompact rotation groups

M. Böhm, Georg Junker

Applications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered. Analysis of the symmetry of the Lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group. Identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold. The integration is performed using the orthogonality of the representations. Compact and noncompact rotation groups are considered, where the corresponding path integral is embedded in Euclidean and pseudo-Euclidean spaces, respectively. The unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient. In the first case Fourier analysis leads to an expansion in group characters. In the second case an expansion in zonal spherical functions is obtained. As examples the groups SO(n), SU(2), SO(n−1,1), and SU(1,1) are explicitly discussed. The path integral on SO(n+m) and SO(n,m) in bispherical coordinates is also treated.

Journal of Mathematical Physics•Sep 1, 1987•93 citations

On the path integration for the potential barrier $V_{0}\cosh ^{-2}(ωx)$

Larbi Guechi, Théophile F. Hammann

The propagator associated to the potential barrier $V=V_{0}\cosh ^{-2}(ωx)$ is obtained by solving path integrals. The method of delta functionals based on canonical and other transformations is used to reduce the path integral for this potential into a path integral for the Morse potential problem. The dimensional extension technique is seen to be essential for performing the multiple integral representation of the propagator. The correctly normalized scattering wave functions and the scattering function are derived. To test the method employed, the free particle and the $δ-$function barrier are considered as limiting cases.

arXiv:quant-ph/0302009•NCimB•Feb 2, 2003•0 citations

Path integration via summation of perturbation expansions and applications to totally reflecting boundaries, and potential steps

Ch. Grosche

The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of \ensuremath{\delta}-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the \ensuremath{\delta}-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.

arXiv:hep-th/9303128•Physical Review Letters•Jul 5, 1993•83 citations

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

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

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.

Journal of Physics A Mathematical and General•Jun 7, 2001•17 citations

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
Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and non-compact bases, and new addition theorems are derived which are required in path integral applications involving a positively signed potential. The Green function for the potential barrier is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup. The transition and the reflection coefficients are given. Results for the moving barrier are also presented.
Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and noncompact basis and the new addition theorems are derived which are required in path integral applications involving the positively signed potential. The Green function for the potential barrier V=cosh−2omegaxV=cosh^{-2}\\omega xV=cosh−2omegax is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup.The transition and the reflection coefficients are given.Results for the moving barrier V=cosh−2omega(x−g0t)V=cosh^{-2}\\omega (x-g_0t)V=cosh−2omega(x−g0​t) are also presented.
Applications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered. Analysis of the symmetry of the Lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group. Identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold. The integration is performed using the orthogonality of the representations. Compact and noncompact rotation groups are considered, where the corresponding path integral is embedded in Euclidean and pseudo-Euclidean spaces, respectively. The unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient. In the first case Fourier analysis leads to an expansion in group characters. In the second case an expansion in zonal spherical functions is obtained. As examples the groups SO(n), SU(2), SO(n−1,1), and SU(1,1) are explicitly discussed. The path integral on SO(n+m) and SO(n,m) in bispherical coordinates is also treated.
On the path integration for the potential barrier V0cosh⁡−2(ωx)V_{0}\cosh ^{-2}(ωx)V0​cosh−2(ωx)
The propagator associated to the potential barrier V=V0cosh⁡−2(ωx)V=V_{0}\cosh ^{-2}(ωx)V=V0​cosh−2(ωx) is obtained by solving path integrals. The method of delta functionals based on canonical and other transformations is used to reduce the path integral for this potential into a path integral for the Morse potential problem. The dimensional extension technique is seen to be essential for performing the multiple integral representation of the propagator. The correctly normalized scattering wave functions and the scattering function are derived. To test the method employed, the free particle and the δ−δ-δ−function barrier are considered as limiting cases.
The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of \ensuremath{\delta}-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the \ensuremath{\delta}-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.
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.