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Physical Review A

Exact Green’s function for the step and square-barrier potentials

Marcus A. M. de Aguiar

Published Oct 1, 1993• 33 citations
Publisher

Abstract

We calculate both the exact retarded and advanced Green's function for the one-dimensional step and square-barrier potentials in the space-energy representation. Some of the results for the square barrier are also extended to general symmetric potentials of finite range.
DOI
10.1103/physreva.48.2567

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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.
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.