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American Journal of Physics

Wronskian method for one-dimensional quantum scattering

Francisco Marcelo Fernández

Published Jul 26, 2011• 15 citations
Publisher

Abstract

The use of Wronskians with properly chosen linearly independent solutions of the Schrödinger equation greatly facilitates the calculation of the transmission probability for scattering in one dimension and also provides a reliable test for the accuracy of the calculation. We apply the approach to a Gaussian barrier and a Gaussian well, and test its accuracy for an exactly solvable model.

DOI
10.1119/1.3596393

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The use of Wronskians with properly chosen linearly independent solutions of the Schrödinger equation greatly facilitates the calculation of the transmission probability for scattering in one dimension and also provides a reliable test for the accuracy of the calculation. We apply the approach to a Gaussian barrier and a Gaussian well, and test its accuracy for an exactly solvable model.
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.
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.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. The 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. The 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. The chapter concludes with a consideration of identical particle scattering and an application to the proton–proton case, where a dramatic manifestation of the underlying substructure emerges at high energies.