Scattering Theory of Waves and Particles pp 121-140 | Cite as

# Particle Scattering in Classical Mechanics

Chapter

## Abstract

The scattering of classical particles by a central field of force or by one another is described in terms of the particle’s orbit of motion. The latter is most simply obtained from the hamiltonian via the Hamilton-Jacobi equation.

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### Notes and References

- 5.1For discussions of the Hamilton-Jacobi equation see, for example, H. Goldstein, (1980), where further references can be found at the end of chaps. 9 and 10.Google Scholar
- 5.3The cross section for the scattering of charged particles in a Coulomb field was first calculated by E. Rutherford (1911), and it was instrumental in his discovery of the atomic nucleus from the atomic scattering of
*α*particles. It is one of the striking facts of quantum mechanics that, if effects of relativity are neglected, this cross section has exactly the same value when calculated classically and quantum mechanically. We can only speculate for how long the discovery of the nucleus would have been delayed if it had not been for this coincidence which made the classical argument used by Rutherford correct.Google Scholar - For a detailed explanation of a general contour integration technique of carrying out an integral such as (5.6), say, in the Coulomb case, see A. Sommerfeld (1933), pp. 645ff.Google Scholar
- See I. Herbst (1974a) for trajectories of particles in a Coulomb field.Google Scholar
- 5.4For discussions of orbiting, or spiral scattering, see Hirschfelder, Curtiss, and Bird, (1954) and Ford and Wheeler (1959).Google Scholar
- 5.5For discussions of glory and rainbow scattering in electromagnetic scattering, see Chap. 3; for their discussion in classical particle scattering, see Ford and Wheeler (1959).Google Scholar
- 5.6A discussion of classical particle scattering by an r
^{-4}potential can be found in Vogt and Wannier (1954).Google Scholar - 5.7The derivation of the relation between the c.m. cross section and the cross section in a general coordinate system follows R. G. Newton (1972). For a detailed discussion of classical scattering in the laboratory system, see M. Gryzinski (1965a).Google Scholar
- 5.9The content of this section is based on a paper by Keller, Kay, and Shmoys (1956), in which the examples of Rutherford scattering and scattering by an
*r*^{-2}potential are also worked out explicitly. See also F. C. Hoyt (1939); Minerbo and Levy (1969); and W. H.Miller (1969b).Google Scholar - 5.9aIntegral equations of the Abel type are given in G. Doetsch (1956), pp. 157ff. Integrals of the Euler-transform type (5.41) are also called
*Riemann-Liouville fractional integrals*, and they are tabulated in A. Erdelyi (1954), vol. 2, pp. 185ff.Google Scholar - 5.9b
*General Note.*For formulations of classical particle scattering in a language related to that used in quantum mechanics, see Prigogine and Henin (1957 and 1959); P. Résibois (1959); Miles and Dahler (1970); B. C. Eu (1971a); W. Hunziker (1974); T. A. Osborn et al. (1980); Narnhofer and Thirring (1981); Bollé and Osborn (1981); K. Yajima(1981e).Google Scholar - 5.9cFor classical treatments of the three-body scattering problem, in which particle
*A*impinges upon a bound system of particles*B*and C and either*A*and the bound system (2?,*C)*emerge unscathed or else*B*and the bound system*(A*, C), or Cand*(A, B)*, emerge, see Karplus, Porter, and Sharma (1964 and 1965), Karplus and Raff (1964), and Biais and Bunker (1964). See also M. Gryzinski (1965b); B. C. Eu (1971).Google Scholar

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