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Prog. Theor. Phys. Vol. 39 No. 2 (1968) pp. 430-452

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An Impact Parameter Representation of the Scattering Problem*

Toshimi Adachi and Tsuneyuki Kotani*

The Tokyo Metropolitan Technical College, Sinagawa, Tokyo
*Department of Physics, Tokyo Metropolitan University, Setagaya, Tokyo

(Received August 17, 1967)

Abstract:

The new exact treatment of the scattering problem proposed by Adachi and Kotani is summarized in a more complete form. This impact parameter representation and the partial wave expression are complementary. The former is more suitable in the short wavelength region, while the latter in the long wavelength region. By assuming a short wavelength approximation only, many formulae are simpliried in the former expansion; in the case of the potential scattering, the simple relation between the scattering amplitude and the (optical) potential is obtained by applying this approximation to the correct solution in the complete form of the Born series. This result is identical with the form obtained under the eikonal or semiclassical approximation. It is proved that the total cross section in the high energy region is expressed accurately in this approximate form, but the angular distribution is less accurate for some cases. As examples, the scattering by a square well potential and the charge exchange scattering of protons by hydrogen atoms are discussed. The relations with the other impact parameter formalisms proposed by Blankenbecler and Goldberger and others are discussed. The solution obtained by Blankenbecler and Goldberger for potential scattering is shown to be the simplest solution which satisfies the unitarity relation under the short wavelength approximation. The reason why their solution is different from the eikonal form is explored by deriving the exact dispersion relation. The Schrödinger equation in the impact parameter representation is derived.


URL : http://ptp.ipap.jp/link?PTP/39/430/
DOI : 10.1143/PTP.39.430


* For an editorial reasen, this paper was separated into two parts. The second part is published in Vol. 39 No. 3. (See Ref. 1.)

[ Full Text PDF : FREE ACCESS (1196K) ] Citation:

References:

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Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 39 No. 3 (1968) pp. 785-816 :
    An Impact Parameter Representation of the Scattering Problem*
    Toshimi Adachi and Tsuneyuki Kotani
  2. Progress of Theoretical Physics Vol. 39 No. 4 (1968) pp. 1006-1019 :
    An Impact-Parameter Representation of Pion-Nucleon Scattering Amplitudes at High Energies
    Reijiro Kubo
  3. Progress of Theoretical Physics Vol. 44 No. 2 (1970) pp. 562-564 :
    Complex Angular Momentum and Impact Parameter Amplitude
    Toshimi Adachi
  4. Progress of Theoretical Physics Vol. 44 No. 5 (1970) pp. 1425-1427 :
    The Impact Parameter Amplitude with a Direct-Channel Regge Pole
    Toshimi Adachi
  5. Progress of Theoretical Physics Vol. 49 No. 1 (1973) pp. 242-255 :
    Relativistic Eikonal Approximation and Vector Meson Exchange Effect
    Masaru Doi and Tsuneyuki Kotani
  6. Progress of Theoretical Physics Vol. 49 No. 2 (1973) pp. 586-593 :
    Analytic Properties of the Impact Parameter Amplitude
    Takashi Obara
  7. Progress of Theoretical Physics Vol. 60 No. 5 (1978) pp. 1602-1603 :
    Relation between Eikonal Multiple Scattering Expansion and Born Series
    Toshimi Adachi, Toshio Inagaki and Gyo Sakaguchi

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