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Multiple Reflections in One-Dimensional Quantum Scattering

Wolf Jung

Published Jan 1, 2002• 1 citation
OpenAlex

Abstract

Consider the one-dimensional scattering of an electron from two potentials with non-overlapping supports. We show that the scattering amplitudes can be obtained from those of the single potentials, and these relations can be explained in terms of multiple reflections of partial waves in the region between the two potentials. From http://www.iram.rwth-aachen.de/~jung/, the latest version of this paper is available as jg-97-1.*, and from http://www.ma.utexas.edu/mp arc/, the first version is obtained as 97-659.latex. 1 Introduction The scattering of light by a parallel sheet of glass can be described in terms of the scattering by the front and back surfaces. We show that a similar description is possible for Schrodinger scattering with potentials V 1;2 of compact support, where supp V 1 ae [a; b] is located left of supp V 2 ae [c; d]. Thus the scattering by V = V 1 + V 2 can be understood as a result of multiple reflections between the potentials V 1 and V 2 . The scattering amplitudes...

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Consider the one-dimensional scattering of an electron from two potentials with non-overlapping supports. We show that the scattering amplitudes can be obtained from those of the single potentials, and these relations can be explained in terms of multiple reflections of partial waves in the region between the two potentials. From http://www.iram.rwth-aachen.de/~jung/, the latest version of this paper is available as jg-97-1.*, and from http://www.ma.utexas.edu/mp arc/, the first version is obtained as 97-659.latex. 1 Introduction The scattering of light by a parallel sheet of glass can be described in terms of the scattering by the front and back surfaces. We show that a similar description is possible for Schrodinger scattering with potentials V 1;2 of compact support, where supp V 1 ae [a; b] is located left of supp V 2 ae [c; d]. Thus the scattering by V = V 1 + V 2 can be understood as a result of multiple reflections between the potentials V 1 and V 2 . The scattering amplitudes...
One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrodinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in the partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from locality of the potential in the wave equation and represent the effect of causality in time dependence of the scattering process.
One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrödinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from the locality of the potential in the wave equation and represent the effect of causality in the time dependence of the scattering process.
We study the partial-wave description of scattering of a particle from a one-dimensional potential in quantum mechanics. There are two partial waves, with even and odd parity. They are decoupled if the potential V(x) is symmetric, i.e., V(x)=V(−x). If V(x)≠V(−x), however, the two partial waves are coupled. This coupled-channel problem can be handled in terms of two eigenphases and a mixing parameter. This method enables us to shed new light on the transmission and reflection probabilities at threshold.
A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. ƒe integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. ƒe relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. ƒe chapter concludes with a consideration of identical particle scattering and an application to the protonproton case, where a dramatic manifestation of the underlying substructure emerges at high energies.