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Potentials from the Scattering Amplitude at Fixed Energy: General Equation and Mathematical Tools

  • K. Chadan
  • P. C. Sabatier
  • R. G. Newton
Chapter
Part of the Texts and Monographs in Physics book series (TMP)

Abstract

In the following three chapters we only deal with problems in which spherical symmetry holds. Now, let us assume that the scattering amplitude F(k, cos θ) has been constructed from the cross section. The next step in studying the interaction is to introduce a mathematical model in which the interaction appears as a parameter, and to determine that parameter from F(k, cos θ). Such is the Schrödinger equation with a spherically symmetric potential V(r) in the case of two colliding particles, below the first inelastic threshold, in the nonrelativistic range. In detail we have
$${\nabla ^2}\Psi + [{K^2} - V(r)]\Psi = 0,$$
where r is the distance between the two particles, k2 andV(r) are, respectively, proportional to the energy and the potential energy in terms of the reduced mass, k2 = 2mE/h2, V(r) = 2mV(r)/h2.

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Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • K. Chadan
    • 1
  • P. C. Sabatier
    • 2
  • R. G. Newton
    • 3
  1. 1.Laboratoire de Physique ThéoriqueUniversité de Paris-SudOrsayFrance
  2. 2.Université des Sciences et Techniques du LanguedocMontpellier CedexFrance
  3. 3.Department of PhysicsIndiana UniversityBloomingtonUSA

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