Abstract
Having discussed the scattering of classical electromagnetic waves in Part I and that of classical particles in Part II, we now turn to the treatment of particle scattering from the quantum-mechanical point of view. The language, the general approach, and many of the tools here resemble those used in the previous chapters. That is in the nature of quantum mechanics. The description is most easily “visualized” in terms of waves. It therefore bears a great deal of resemblance to that of classical electromagnetic theory. Yet the wave formalism is tied intimately to a particle interpretation, in which language it is related to the discussion of Chap. 5.
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Notes and References
The correspondence between the motion of classical particles and that of quantum-mechanical wave packets is most directly contained in Ehrenfest’s theorem, P. Ehrenfest (1927), which may be found in most standard books on quantum mechanics, e.g., L. I. Schiff (1968), p. 28. For some generalizations, see B. A. Lippmann (1965); E. Gerjuoy (1965c).
The “classical” papers on time-dependent scattering theory are Lippmann and Schwinger (1950); Gell-Mann and Goldberger (1953). For an excellent general review see Brenig and Haag (1959), a translation of which can be found in M. Ross (1963a). This book also contains reprints of the papers by Lippmann and Schwinger and Gell-Mann and Goldberger. Other general reviews are B. S. DeWitt (1955a); F. E. Low (1959) and R.Haag (1961).
Other general references are Jauch and Rohrlich (1955), chaps. 7 and 8; S. S. Schweber (1961), chap. 11; Wu and Ohmura (1962), chap. 4; LaVita and Marchand (1974); J. A. DeSanto et al. (1980).
The following papers may also be consulted: S. T. Ma (1953); Coester, Hamermesh, and Tanaka (1954); S. Sunakawa (1955); H. Eckstein (1956a and b); L. van Hove (1955 and 1956); Jauch and Zinnes (1959).
A specific discussion of the scattering of wave packets is given by Dodd and McCarthy (1964).
For extensions to the relativistic region, see Fong and Sucher (1964); Jordan, Macfarlane, and Sudarshan (1964); F. Coester (1965 and 1980); L. B. Redei (1965); K. Yajima (1976).
On the connection between time-dependent and time-independent theory, see Belinfante and Moller (1953).
The original references to the M011er wave operator are C. M011er (1945 and 1946), the first of which is reprinted in M. Ross (1963a). These papers are important early references on modern scattering theory. For other work on the wave operator, see S. Fubini (1952a); H. Eckstein (1954); M. N. Hack (1958); T. Kato (1967b); K. Yajima (1977); V. Enss (1978). For definitions of the wave operator in the presence of static electric and magnetic fields see Avron and Herbst (1977 and 1978); K. Yajima (1979b and 1981d).
The S matrix was independently invented by Wheeler and Heisenberg, in J. A. Wheeler (1937a); W. Heisenberg (1943 and 1944). For its early use in field theory see E. C. G. Stueckelberg (1943, 1945, and 1946); J. Schwinger (1948); R. P. Feynman (1949); F. J. Dyson (1949). See also the papers by C. Miller (1945 and 1946) and, for a demonstration of the equivalence of various S matrix definitions, S. Fubini (1952).
The interaction picture is closely related to Dirac’s time-dependent amplitudes in the method of variation of constants, introduced by P. A. M. Dirac (1926 and 1927). It has become extremely important in the formulation of relativistically covariant quantum field theory and was introduced for that purpose by S. Tomonaga (1946); J. Schwinger (1948b). For the time-ordering device used in formal expressions for the wave operator and the S operator, see F. J. Dyson (1949); R. P. Feynman (1951); I. Fujiwara(1952).
E. A. Remler (1975) discusses scattering in the Wigner representation.
The scattering-into-cones theorem is due to J. D. Dollard (1969 and 1973); see also J. M. Jauch et al. (1972); J.-M. Combes et al. (1975).
Our attitude in this section, it will be noticed, is not to prove the results of the previous sections rigorously, starting from a given class of hamiltonians, but to point out what notions and distinctions must be introduced for a rigorous treatment and what, exactly, must be shown for the general theory to work. For rigorous demonstrations the reader is referred especially to J. Cook (1957). Other articles using rigorous mathematical treatments of scattering theory from a time-dependent point of view are K. O. Friedrichs (1952); J. M. Jauch (1958a); Kato and Kuroda (1959); A. Galindo Tixaire (1959); S. T. Kuroda (1959a and 1962); Green and Lanford (1960); T. F. Jordan (1962); I. V. Stankevich (1962); F. H. Brownell (1962); J. D. Dollard (1964); R. T. Prosser (1964); F. Rhys (1965); C. H. Wilcox (1965); J.-M. Combes (1980a). The books by B. Simon (1971), W. O. Amrein et al. (1977), and Reed and Simon (1979) contain a wealth of information.
For further references concerning mathematical approaches, see Sec. 7.3.
The following papers may be consulted concerning the use of the adiabatic theorem in scattering theory: H. S. Snyder (1951); B. Ferretti (1951); Suura, Mimura, and Kimura (1952); H. E. Moses (1955).
For some rigorous treatments of the stationary-phase method, see J. G. van der Corput (1934 and 1936); A. Erdelyi (1955); G. Braun (1956).
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Newton, R.G. (1982). Time-Dependent 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_6
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