• Roger G. Newton
Part of the Texts and Monographs in Physics book series (TMP)


In this chapter we wish to consider a number of illustrative examples for which some of the general results discussed previously can be derived or studied directly. What we are particularly interested in is potentials for which the Schrödinger equation can be solved explicitly in terms of known functions. There is a large class of these for which the solution of (11.8) is known for a fixed value of l and all values of the energy. But there are essentially only four kinds of known potentials for which the radial Schrödinger equation can be solved explicitly for all l and all E; these are the square well, the harmonic oscillator well, the Coulomb potential, and the r-4 potential. It would be very desirable for many purposes to know more such potentials. Of course, the Schrödinger equation can always be solved numerically, if necessary on an electronic computer. But for the purpose of investigating the properties of the scattering amplitude it is often very useful to have explicitly solvable examples. What is more, these can be used as starting points for perturbation calculations.


Born Approximation Schrodinger Equation Virtual State Coulomb Field Asymptotic Tail 
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Notes and References

  1. 14.1
    The zero-range potential was first used by Bethe and Peierls (1935).Google Scholar
  2. 14.3
    The exponential potential was first employed and the Schrödinger equation solved for it by Bethe and Bacher (1936), particularly p. 110. See also R. Jost (1947). The general demonstration that potentials with an exponential tail e~ r/a usually lead to poles at k = inßa, «=1,2,..., was given in R. E. Peierls (1959). For a discussion of potentials which are finite sums of exponentials, see A. Martin (1959); see also M. G. Fuda (1971a). For a generalization to the Klein-Gordon equation, see Bawin and Lavině (1974).Google Scholar
  3. 14.3a
    For some numerical values of low-energy phase shifts and a comparison of various approximations for exponential, Yukawa, and gaussian potentials, see L. Wojtczak (1963).Google Scholar
  4. 14.4
    The potentials (14.14) were first used and the Schrödinger equation solved with them by C. Eckart (1930). The special case of the potential (14.13) was introduced by L. Hulthén (1942). See also Jost and Pais (1951); M. G. Fuda (1971b).Google Scholar
  5. 14.5
    The method of this section is due to A. Martin (1959). Its extension to / > 0 is given in A. Martin (1960) and by a different technique in Fivel and Klein (1960).Google Scholar
  6. 14.6
    The Rutherford scattering formula was first derived quantum mechanically in the Born approximation by taking the limit of a screened Coulomb field by G. Wentzel (1926b) and subsequently without screening by J. R. Oppenheimer (1927). The first exact solution was given by N. F. Mott (1928), and both in the partial-wave expansion and with parabolic coordinates, by W. Gordon (1928).Google Scholar
  7. 14.6a
    The symmetrized scattering formula (14.32) is due to N. F: Mott (1930). For discussions of higher Born approximations in the Coulomb case, see R.-H. Dalitz (1951); and Gesztesy and Thaller (1981). The Gamow factor was introduced by G. Gamow (1928); see also Gurney and Condon (1928). For very detailed discussions of Coulomb wave functions see Bethe and Salpeter (1957); Hull and Breit (1959); Curtis (1964); Luk’yanov, Teplov, and Akimova (1965).Google Scholar
  8. 14.6b
    For treatments of the Coulomb Green’s function, see Wichmann and Woo (1961); Hostler and Pratt (1963); L. Hostler (1964); J. Schwinger (1964); and, in the relativistic case, M. E. Rose (1961); Swam and Biedenharn (1963); V. G. Gorshkov (1964); Fradkin, Weber, and Hammer (1964). See also J. H. Hetherington (1963).Google Scholar
  9. 14.6c
    An interesting discussion of the pole distribution of the S matrix for a screened Coulomb field, in the limit of large screening radius, was given by Ferreira and Teixeira (1966). They found that this limiting pole distribution does not coincide with that of the Coulomb S matrix given by (14.42). For other discussions of the screened Coulomb potential, see W. F. Ford (1964); Holdeman and Thaler (1965); W. F. Ford (1966); J. D. Dollard (1968); Prugovečki and Zorbas (1973a); J. R. Taylor (1974); J. A. Zorbas (1974a); E. A. Bartnik et al. (1975); Goodmanson and Taylor (1980).Google Scholar
  10. 14.6d
    The time-dependent scattering theory was extended to the Coulomb case by J. D. Dollard(1964).Google Scholar
  11. 14.6e
    Relativistic Coulomb scattering was discussed, for example, by V. G. Gorshkov (1961); Gluckstern and Lin (1964); Dollard and Velo (1966).Google Scholar
  12. 14.6f
    Detailed studies of analyticity properties of partial-wave scattering amplitudes for charged particles are to be found in Cornille and Martin (1962); Yu. L. Mentkovsky (1962, 1963, 1964, and 1965).Google Scholar
  13. 14.6g
    For further work on scattering by pure and admixed Coulomb potentials, see L. A. Sakhnovich (1965 and 1972); P. Swan (1967a); G. B. West (1967); D. Zwanziger (1967); Mulherin and Zinnes (1970); A. M. Veselova (1970a); J. D. Dollard (1971 and 1972);Google Scholar
  14. 14.6h
    Nuttall and Stagat (1971); W. W. Zachary (1971); S. Ali et al. (1972); Marchesin and O’Carroll (1972); L. Marquez (1972); I. H. Sloan (1973); Chandierand Gibson (1974); I. Herbst (1974b); M. C. Li (1974); P. Hillion (1975); Semon and Taylor (1975); van Haeringen and van Wageningen (1975); H. van Haeringen (1976, 1977, 1978b, 1979a and b); Prugovečki and Zorbas (1978); M. J. Seaton (1978); Banerjee and Chakravorty (1978); J. Zorbas (1977 and 1978); Y. Saito (1979 and 1980a); C. Chandler (1980); F. W. Gesztesy et al. (1980); Gesztesy and Lang (1981); van Haeringen and Kok (1981a and b); Kok and van Haeringen (1981b); S. H. Patil (1981); Coulomb interference is discussed by P. R. Auvil (1968); and V. Franco (1973). See also references for Sec. 12.4. For other extensions, see Luming and Predazzi (1966).Google Scholar
  15. 1.4.7
    The treatment of this section is a generalization ofthat of W. R. Theis (1956). The Bargmann potentials were first described (in a more restricted form) in V. Bargmann (1949b). The special cases described in 14.7.2 were also discussed there. The potential (14.85) was discussed by K. Chadan (1955 and 1956); and by R. G. Newton (1957), as a simple model for the deuteron. In the same context, see M. Blažek (1962a). For a discussion of pathological cases such as (14.87) and others, see Moses and Tuan (1959). For generalizations to coupled systems (spin 1/2 with spin 1/2), see Fulton and Newton (1956); and to coupled channel equations, J. R. Cox (1964); Cox and Garcia (1975); E. B. Plekhanovet al. (1981); B. N. Zakhariev et al. (1981).Google Scholar


  1. For other potentials for which the Schrödinger equation is explicitly solvable, see Vogt and Wannier (1954); R. M. Spector (1964). They discuss the r‒4 potential, which leads to a solvable equation for all k and all l. Much larger classes of potentials are treated by Bhattacharjie and Sudarshan (1962); A. K. Bose (1964); Aly and Spector (1965).Google Scholar

Copyright information

© Springer Science+Business Media New York 1982

Authors and Affiliations

  • Roger G. Newton
    • 1
  1. 1.Department of PhysicsIndiana UniversityBloomingtonUSA

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