Prog. Theor. Phys. Supplement Extra Number (1965) pp. 316-331
An Impact Parameter Formalism
The Tokyo Metropolitan Technical College, Sinagawa, Tokyo
*Department of Physics, Tokyo Metropolitan University, Setagaya, Tokyo
(Received June 25, 1965)
A relativistic impact-parameter formalism is introduced without any approximation. It is applicable for all physical values of energy and scattering angle. An impact parameter amplitude is defined from an invariant scattering amplitude by the integral with the Bessel function, in place of the Legendre function in the definition of a partial-wave amplitude.
We find that this impact parameter amplitude is an example of a new integral expansion. It is proved that one of the necessary and sufficient conditions that this new expansion is permissible, is that the amplitude should satisfy the Kapteyn equation, which is required in the Webb-Kapteyn theory of the Neumann series expansion of an odd function.
The physical interpretation of the amplitude is discussed and cross sections are expressed by simple integrals over the impact parameter without any approximation. The unitarity relation for the amplitude is expressed in the simpler form. Comparison is made with three other relativistic formalisms proposed by Blankenbecler and Goldberger, Ida, and Cottingham and Peiels. In contrast with our formalism, the applications of these three formalisms are limited either to the extremely high energy region or to only the forward angle scattering.
DOI : 10.1143/PTPS.E65.316
- See, for example, R. J. Glauber, Lectures in Theoretical Physics (Interscience publishers Inc., New York, 1958), Vol. 1, p. 315.
R. Blankenbecler and M. Goldberger, Phys. Rev. 126 (1962), 766[APS].
- M. Ida, Prog. Theor. Phys. 28 (1962), 943[PTP]; ibid. 28 (1962), 945[PTP]; The unpublished draft, KUNS-3 (Kyoto University, 1962).
W. N. Cottingham and R. F. Peierls, Phys. Rev. 137 (1965), B147[APS].
R. Omnes, Phys. Rev. 137 (1965), B653[APS].
- E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948) 2nd ed., p. 352.
- G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, 1922), first edition.
- P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Co., New York, 1953), p. 943.
- L. van Hove, Nuovo Cim. 28 (1963), 798;
Rev. Mod. Phys. 36 (1964), 655[APS].
- Bateman Manuscript Project, Higher Transcendental Functions, edited by H. Erdélyi (McGraw-Hill Book Co. Inc., New York, 1953), Vol. 2.
- Bateman Manuscript Project, Tables of Integral Transformations, edited by H. Erdélyi (McGraw-Hill Book Co. Inc., New York, 1953), Vol. 2.
- T. M. MacRobert, Proc. R. Soc. Edinburgh 51 (1931), 116.
- G. H. Hardy and E. C. Titchmarsh, Proc. London Math. Soc. 23 (1924), 1.
D. B. Sears and E. C. Titchmarsh, Quart. J. of Math. (Oxford) 1 (1950), 165.
See also p.355 of reference 6).