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Effective Potential Model for Calculating Nuclear Corrections to the Energy Levels of Hydrogen

H. GROTCH* and D. R. YENNIE

H. GROTCH*

  • Laboratory of Nuclear Studies, Cornell University, Ithaca, New York and Department of Physics, Pennsylvania State University, University Park, Pennsylvania

D. R. YENNIE

  • Laboratory of Nuclear Studies, Cornell University, Ithaca, New York

  • *Present address.
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Rev. Mod. Phys. 41, 350 – Published 1 April, 1969

DOI: https://doi.org/10.1103/RevModPhys.41.350

Abstract

The present work is an attempt to re-evaluate the nuclear corrections to the energy levels of hydrogen by using an effective potential approach. The basic idea is to infer from electron-proton scattering a potential which may then be applied to the bound-state problem. In lowest order, the potential is chosen from the first-order Feynman diagram for the scattering. With this choice the Breit equation is obtained. It is then solved in an approximate way, in the non-relativistic limit of the proton, to obtain wave functions which are accurate enough for use in evaluating the effects of perturbations of the potential. The reduced mass corrections to the fine structure and the hyperfine structure levels are readily found. The effect on the hyperfine splitting of the distribution of the proton charge and magnetic moment is obtained by correcting the lowest-order potential to include the proton form factors. A further modification is needed in evaluating additional recoil corrections, of relative order αmM, to the fs and the hfs. This additional term accounts for the failure of the iteration of the lowest-order potential to reproduce the scattering obtained from the second-order Feynman diagrams. The α2mM contribution to the state-dependent mass corrections to the hfs is also analyzed within the context of this approach. All the corrections found are in complete agreement with previous results obtained by the Bethe-Salpeter (BS) equation, but the present method has the virtue of conceptual simplicity.

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References (52)

  1. C. K. Iddings and P. M. Platzman, Phys. Rev. 113, 192 (1959)
  2. S. B. Crampton, D. Kleppner, and N. F. Ramsey, Phys. Rev. Letters 11, 338 (1963)
  3. G. W. Erickson and D. R. Yennie, Ann. Phys. (N.Y.) 35, 271, 447 (1965)
  4. M. F. Soto, Jr., Phys. Rev. Letters 17, 1153 (1966)
  5. S. Triebwasser, E. S. Dayhoff, and W. E. Lamb, Phys. Rev. 89, 98 (1953)
  6. R. T. Robiscoe, Phys. Rev. 138, A22 (1965) R. T. Robiscoe and B. L. Cosens, Phys. Rev. Letters 17, 69 (1966)
  7. E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951)
  8. M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951)
  9. W. E. Cleland, J. M. Bailey, M. E. Eckhause, V. W. Hughes, R. M. Mobley, R. Prepost, and J. E. Rothberg, Phys. Rev. Letters 13, 202 (1964)
  10. W. H. Parker, B. N. Taylor, and D. N. Langenburg, Phys. Rev. Letters 18, 287 (1967)
  11. H. Grotch and D. R. Yennie, Z. Physik 202, 425 (1967)
  12. A. A. Logunov and Tavkhelidze, Nuovo Cimento 29, 380 (1963)
  13. R. N. Faustov, Nucl. Phys. 75, 669 (1966)
  14. S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper and Row, Publishers, Inc., New York, 1962), 2nd ed., p. 315
  15. G. Breit, Phys. Rev. 34, 553 (1929) G. Breit and G. E. Brown, ibid. 74, 1278 (1948)
  16. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957)
  17. L. H. Chan, K. W. Chen, J. R. Dunning, Jr., N. F. Ramsey, J. K. Walker, and R. Wilson, Phys. Rev. 141, 1298 (1966)
  18. E. Fermi, Z. Physik 60, 320 (1930)
  19. G. Breit and R. E. Meyerott, Phys. Rev. 72, 1023 (1947) G. Breit, G. E. Brown, and G. B. Arfken, ibid. 76, 1299 (1949)
  20. N. Kroll and F. Pollock, Phys. Rev. 84, 594 (1951) ibid.86, 876 (1952)
  21. R. Karplus, A. Klein, and J. Schwinger, Phys. Rev. 84, 597 (1951)
  22. A. J. Layzer, Bull. Am. Phys. Soc. 6, 514 (1961) Nuovo Cimento 33, 1538 (1964)
  23. D. W. Zwanziger, Bull. Am. Phys. Soc. 6, 514 (1961) Nuovo Cimento 34, 77 (1964)
  24. S. J. Brodsky and G. W. Erickson, Phys. Rev. 148, 26 (1966)
  25. A. C. Zemach, Phys. Rev. 104, 1771 (1956)
  26. G. E. Brown and G. B. Arfken, Phys. Rev. 76, 1305 (1949)
  27. H. Grotch, Ph.D. thesis, Cornell University, 1967 (unpublished)
  28. E. E. Salpeter, Phys. Rev. 87, 328 (1952)
  29. R. Arnowitt, Phys. Rev. 92, 1002 (1953)
  30. W. A. Newcomb and E. E. Salpeter, Phys. Rev. 97, 1146 (1955)
  31. R. Karplus and A. Klein, Phys. Rev. 85, 972 (1952) R. Karplus, A. Klein, and J. Schwinger, ibid. 86, 288 (1952)
  32. C. K. Iddings, Phys. Rev. 138, B446 (1965)
  33. F. Guerin, Nuovo Cimento 50A, 1, 211 (1967)
  34. C. K. Iddings and P. M. Platzman, Phys. Rev. 115, 919 (1959)
  35. A. Verganelakis and D. Zwanziger, Nuovo Cimento 39, 613 (1965)
  36. S. D. Drell and J. D. Sullivan, Phys. Rev. 154, 1477 (1967)
  37. T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954)
  38. J. Schwinger, Proc. Natl. Acad. Sci. U.S. 37, 452, 455 (1951)
  39. R. Karplus and A. Klein, Phys. Rev. 87, 328 (1952)
  40. T. Fulton and R. Karplus, Phys. Rev. 93, 1109 (1954)
  41. H. A. Bethe, Phys. Rev. 72, 339 (1947)
  42. H. A. Bethe, L. M. Brown, and J. R. Stehn, Phys. Rev. 77, 370 (1950) J. M. Harriman, ibid. 101, 594 (1956) C. Schwartz and J. J. Tiemann, Ann. Phys. (N.Y.) 6, 178 (1959)
  43. I. M. Ryshik and I. S. Gradstein, Tables of Series, Products and Integrals (Plenum Press, Inc., New York, 1963), 2nd ed
  44. F. Gross (private communication)
  45. W. A. Barker and F. N. Glover, Phys. Rev. 99, 317 (1955)
  46. M. M. Sternheim, Phys. Rev. 130, 211 (1963)
  47. L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950)
  48. C. Schwartz, Ann. Phys. (N.Y.) 6, 156 (1959)
  49. M. M. Sternheim, Phys. Rev. 128, 676 (1962)
  50. E. S. Dayhoff, S. Triebwasser, and W. E. Lamb, Jr., Phys. Rev. 89, 106 (1953)
  51. D. R. Yennie, Proc. Intern. Symp. Electron Photon Interactions High Energ., Stanford, 1967, 32 (1967)
  52. S. J. Brodsky and E. de Rafael, Phys. Rev. 168, 1620 (1968)

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Rev. Mod. Phys. 41, 350– Published 1 April, 1969

    © 1969 American Physical Society

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