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Journal of Physics A Mathematical and Theoretical

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

H. Ahmedov, I. H. Duru

Published Sep 23, 1996• 0 citations
OpenAlex

Abstract

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.

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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.

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arXiv:hep-th/9303128•Physical Review Letters•Jul 5, 1993•83 citations
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.
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.
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.
The problem of stationary scattering is discussed using the Scbroedinger equation for a region containing the scattering potertial and a potertial barrier. The potential barrier is expressed in Cartesian coordinates. The Green function is determined for this problem, and it is shown that its asyxaptotic expression contains the solution of the Schroedinger equation for an unperturbed barrier for an accurnte choice of scattering direction. The absence of glancing scattering along the barrier was established by means of the Green function, and the basis of a perturbation method for the scattering problem is given. Other approximations for this problem are discussed. (tr-auth)
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.