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Physical Review A

Application of the generalized scattering amplitude in quantum potential scatterings

R. T. Ling

Published Jul 1, 1994• 4 citations
Publisher

Abstract

The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr\"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.

DOI
10.1103/physreva.50.435

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The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (e\ensuremath−{\mathit{e}}^{\mathrm{\ensuremath{-}}}e\ensuremath−,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.
The functional integral method can be used in quantum mechanics to find the scattering amplitude for particles in the external field. We will obtain the potential scattering amplitude form the complete Green function in the corresponding external field through solving the Schrodinger equation, after being separated from the poles on the mass shell, which takes the form of an eikonal (Glauber) representation in the high energy region and the small scattering angles. Consider specific external potentials such as the Yukawa or Gaussian potential, we will find the corresponding differential scattering cross-sections.
The functional integral method can be used in quantum mechanics to find the scattering amplitude for particles in the external field. We obtained the potential scattering amplitude from the complete Green function in the corresponding external field through solving the Schrodinger equation, after being separated from the poles on the mass shell, which takes the form of an eikonal (Glauber) representation in the high energy region and the small scattering angles. Considering specific external potentials such as the Yukawa or Gaussian potential, we found the corresponding differential scattering cross-sections.
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.
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.
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.