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ON THE GREEN FUNCTION FOR SCATTERING IN A REGION WITH A POTENTIAL BARRIER

N.B. Aizenberg

Published Jan 1, 1964• 0 citations
OpenAlex

Abstract

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)

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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)
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We obtain a mathematical and algebraic expression of the Green-function method in treating quantum tunnelling in one-dimensional barrier systems. The retarded Green's function of the free-particle resolvent is then obtained consistently to obtain scattering amplitudes using the Lippmann-Schwinger equation. The technique is used to apply to the d-function barrier, the finite square barrier, and the double-barrier quantum-well structure in order to describe the transmission, penetration of the barrier, and resonance. With the Green, the pole structure in the case of the double-barrier directly traces the quasi-bound and the Breit-Wigner-type resonances.
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