Theoretical and Mathematical Physics

, Volume 98, Issue 1, pp 42–54 | Cite as

Discrete symmetries in quantum scattering

  • A. I. Nikishov
Article

Abstract

Using the discrete symmetries of the Klein—Gordon, Dirac, and Schrödinger wave equations, we obtain from one solution, considered as a function of the quantum numbers and the parameters of the potentials, three other solution. Taken together, these solutions form two complete sets of solutions of the wave equation. The coefficients of the linear relations between the functions of these sets — the connection coefficients — are related in a simple manner to the wave transmission and reflection amplitudes. By virtue of the discrete symmetries of the wave equation, the connection coefficients satisfy certain symmetry relations. We show that in a number of simple cases, the behavior of the wave function near the center of formation of an additional wave determines the amplitude of the wave that is formed at infinity.

Keywords

Reflection Wave Function Wave Equation Quantum Number Linear Relation 

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References

  1. 1.
    Willard Miller (Jr.),Symmetry and Separation of Variables, Addison-Wesley (1977).Google Scholar
  2. 2.
    I. A. Malkin and V. I. Man'ko,Dynamical Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).Google Scholar
  3. 3.
    V. I. Fushchich and A. G. Nikishin,Symmetry of the Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1990).Google Scholar
  4. 4.
    A. I. Nikitov,Tr. Fiz. Inst. Akad. Nauk,111 (1979).Google Scholar
  5. 5.
    A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko,Vacuum Quantum Effects in Strong Fields [in Russian], Énergoizdat, Moscow (1988).Google Scholar
  6. 6.
    L. D. Landau and E. M. Lifshitz,Quantum Mechanics: Nonrelativistic Theory, 3rd ed., Pergamon Press, Oxford (1977).Google Scholar
  7. 7.
    A. I. Akhiezer and V. B. Berestetskii,Quantum Electrodynamics, Interscience, New York (1965).Google Scholar
  8. 8.
    A. Messiah,Quantum Mechanics, Vol. II, North-Holland, Amsterdam (1965).Google Scholar
  9. 9.
    A. I. Nikishov,Yad. Fiz.,46, 163 (1987).Google Scholar
  10. 10.
    S. Flügge,Practical Quantum Mechanics, Vols. 1 and 2, (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vols. 177, 178), Springer-Verlag, Berlin (1971).Google Scholar
  11. 11.
    M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover (1964).Google Scholar
  12. 12.
    E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, Cambridge (1965).Google Scholar
  13. 13.
    Y. Luke,Mathematical Functions and Their Approximations, Academic Press, New York (1975).Google Scholar
  14. 14.
    J. C. P. Miller,Tables of Weber Parabolic Cylinder Function, HMSO, London (1955).Google Scholar
  15. 15.
    M. V. Berry,Proc. R. Soc. London, Ser. A,422, 7 (1989).Google Scholar
  16. 16.
    A. I. Nikishov and V. I. Ritus,Teor. Mat. Fiz.,92, 24 (1992).Google Scholar
  17. 17.
    A. Érdelyi et al. (eds.),Higher Transcendental Functions, (California Institute of Technology H. Bateman M.S. Project), Vol. 1, McGraw Hill, New York (1953).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • A. I. Nikishov

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