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arXiv:1101.0957

Quantum scattering by Wronskians

Francisco Marcelo Fernández

Published Jan 5, 2011• 3 citations
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Abstract

We show that Wronskians between properly chosen linearly independent solutions of the Schrödinger equation greatly facilitate the study of quantum scattering in one dimension. They enable one to obtain the necessary relationships between the coefficients that determine the asymptotic behavior of the wavefunction. As illustrative examples we calculate the transmission probability for the penetration of a Gaussian barrier and the scattering resonances for a Gaussian well.

DOI
10.48550/arxiv.1101.0957

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We show that Wronskians between properly chosen linearly independent solutions of the Schrödinger equation greatly facilitate the study of quantum scattering in one dimension. They enable one to obtain the necessary relationships between the coefficients that determine the asymptotic behavior of the wavefunction. As illustrative examples we calculate the transmission probability for the penetration of a Gaussian barrier and the scattering resonances for a Gaussian well.
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We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model, the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.
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.
Elementary numerical methods are used to evaluate solutions to Schrödinger’s equation for a variety of one-dimensional potential barriers and to determine the transmission coefficients of free particles. These techniques are used to study the oscillatory variations in transmission coefficients for both continuous and discontinuous potentials. The amplitude of the variations is found to be strongly increased by discontinuities in the potential function but small variations exist even for a smooth, continuous potential function. The techniques described are suitable for intermediate-level undergraduate physics students.