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Scattering Theory of Waves and Particles pp 175–210Cite as

Time-Independent Formal Scattering Theory

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Abstract

While the time-dependent scattering theory has great mathematical, physical, and conceptual advantages over the theory based on the time-independent Schrödinger equation, for the purpose of actual calculation, at least in the nonrelativistic domain, it is not very useful. Such calculations are most conveniently performed at a fixed energy. It has already been pointed out, however, that many of the mathematical steps involved require the use of wave packets for convergence. In the stationary-state theory wave packets cannot be used without a constant presence of cumbersome weight functions and integrations over the energy. Without them certain convergence difficulties are inevitable.

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Notes and References

  1. For general treatments of scattering theory from a time-independent point of view we give the following references: Mott and Massey (1949); W. Heitler (1954); Jauch and Rohrlich (1955); Wu and Ohmura (1962); W. O. Amrein et al. (1977); Reed and Simon (1979); as well as most modern books on quantum mechanics. Other general surveys of the formal theory are B. S. DeWitt (1955a); Brenig and Haag (1959); F. E. Low (1959); and R. Haag (1961). See also the important papers C. M011er (1945 and 1946).

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  2. Other articles on formal scattering theory from a time-independent point of view are E. Feenberg(1948); M. L. Goldberger (1951a and b); M. N. Hack(1954); S. Epstein (1955); B. S. DeWitt (1955b); H. Rollnik (1956); Foldy and Tobocman (1957); E. Gerjuoy (1958a and b); T. Sasakawa (1963); K. L. Kowalski (1972b); S. K. Adhikari (1979b); as well as the papers by Wheeler, Heisenberg, Lippmann and Schwinger, and Gell-Mann and Goldberger mentioned in Chap. 6. For a discussion of the relation between the time-dependent and time-independent points of view, see Belinfante and M011er(1953).

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  3. For early use of Green’s-function techniques in scattering problems, see A. Sommerfeld (1910, 1912, and 1931); J. Meixner(1933, 1934, and 1937). The following paper makes particular use of the resolvent, or Green’s function: M. Schönberg (1951).

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  4. See the Notes and References for Secs. 6.3 and 6.4.

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  5. For a specific discussion of the occurrence of the incoming wave state as a final state and the outgoing wave state as the initial state, see Breit and Bethe (1954) and also S. Altshuler (1956). For studies of the off-energy-shell T matrix, see J. Nuttall (1967c); R. D. Amado (1970); M. G. Fuda (1971c); and B. R. Karlsson (1972).

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  6. The so-called optical theorem was discovered in quantum mechanics by E. Feenberg (1932). It is sometimes unjustifiably referred to as the Bohr-Peierls-Placzek relation, after Bohr, Peierls, and Placzek (1939). For some remarks concerning its history in electromagnetic theory, where it was first known, see the Notes and References to Sec. 1.3.9. Its history is traced in detail in R. G. Newton (1976a).

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  7. The utility and validity of expanding the T matrix in terms of operators of finite rank is discussed by T. A. Osborn (1969, 1973) and J. S. Levinger (1973). For a specific discussion of the effects of exchange, see R. Mapleton (1954). The Low equation was introduced by F. E. Low (1955). Its solvability is discussed by R. L. Warnock (1968).

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  8. For a standing-wave form of scattering theory, see Kouri and Levin (1974b).

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  9. Equation (7.69) was formulated originally by C. Lovelace (1964). For an extensive discussion of the time-reversal operation, see E. P. Wigner (1959), chap. 26. For discussions of reciprocity and its relation to detailed balance, see, for example, Wigner and Eisenbud (1947); Blatt and Weisskopf (1952), chap. X, 2D and E. See also F. Coester (1953); S. Watanabe (1955).

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  10. For discussion and derivations of (7.76), (7.76a), see Day, Rodberg, Snow, and Sucher (1961); W. R. Gibbs (1974); and J. Zorbas (1976a).

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  11. Equations (7.87) are due to L. D. Faddeev (1960, 1961, and 1962). Equation (7.92) is due to S. Weinberg (1964), sec. III. See also W. Hunziker (1964).

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  12. The following books, among others, may be consulted on the mathematical questions that arise in this chapter: J. von Neumann (1955); Riesz and Sz-Nagy (1955); B. Friedman (1957); Hille and Phillips (1957); A. E. Taylor (1958); Dunford and Schwartz (1958 and 1963); K. O. Friedrichs (1960): Akhiezer and Glazman (1961); K. Yosida (1965); Reed and Simon (1972). Among these, perhaps the books by Taylor and Reed and Simon are the most accessible to nonmathematicians. The following article may also usefully be consulted: S. Weinberg (1964a), especially appendix A.

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  13. To a reader unfamiliar with vector-space theory the following book is highly recommended as a first approach before proceeding to Hilbert-space theory: P. R. Halmos (1942).

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  14. An important paper on the self-adjoint nature of the hamiltonian is T. Kato (1951a); see also N. Limic (1966).

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  15. The following mathematical papers concerned with the change of continuous spectra under perturbations are also relevant: K. O. Friedrichs (1948); T. Kato (1957a and b); M. Rosenblum (1957); N. Aronszajn (1957); Ladyzhenskaya and Faddeev (1958); S. T. Kuroda (1959 and 1960); L. Thomas (1972). See also A. Ya. Povsner (1953 and 1955). For other rigorous treatments of scattering theory from the time-independent point of view see K. Kodaira (1949 and 1950); T. Ikebe (1960); K. Meetz (1962); T. F. Jordan (1962); L. D. Faddeev (1963); J. G. Belinfante (1964); B. Simon (1971); S. Agmon (1975); W. O. Amrein et al. (1977); Reed and Simon (1979); J.-M. Combes (1980b). See also Zemach and Odeh (1960); and B. Misra et al. (1963).

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  16. The remarks connecting a Mittag-Leffler expansion of the resolvent to completeness were stimulated by Fonda et al. (1966).

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  1. Department of Physics, Indiana University, Swain Hall-West 117, 47405, Bloomington, IN, USA

    Roger G. Newton

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© 1982 Springer Science+Business Media New York

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Newton, R.G. (1982). Time-Independent Formal Scattering Theory. In: Scattering Theory of Waves and Particles. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88128-2_7

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  • DOI: https://doi.org/10.1007/978-3-642-88128-2_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-88130-5

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