Skip to main content
SpringerLink
Log in

Hydrogenic Wave Functions

  • Reference work entry
Springer Handbook of Atomic, Molecular, and Optical Physics

Part of the book series: Springer Handbooks ((SHB))

Abstract

This chapter summarizes the solutions of the one-electron nonrelativistic Schrödinger equation, and the one-electron relativistic Dirac equation, for the Coulomb potential. The standard notations and conventions used in the mathematics literature for special functions have been chosen in preference to the notations customarily used in the physics literature whenever there is a conflict. This has been done to facilitate the use of standard reference works such as Abramowitz and Stegun [9.1], the Bateman project [9.2,3], Gradshteyn and Ryzhik [9.4], Jahnke and Emde [9.5], Luke [9.6,7], Magnus, Oberhettinger, and Soni [9.8], Olver [9.9], Szego [9.10], and the new NIST Digital Library of Mathematical Functions project, which is preparing a hardcover update [9.11] of Abramowitz and Stegun [9.1] and an online digital library of mathematical functions [9.12]. The section on special functions contains many of the formulas which are needed to check the results quoted in the previous sections, together with a number of other useful formulas. Itincludes a brief introduction to asymptotic methods.

References to the numerical evaluation of special functions are given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Abbreviations

NIST:

National Institute of Standards and Technology

References

  1. M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions (Dover, New York 1965)

    Google Scholar 

  2. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi: Higher Transcendental Functions, Vol. 1, 2, 3 (McGraw-Hill, New York 1955) p. 1953

    MATH  Google Scholar 

  3. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi: Tables of Integral Transforms, Vol. 1, 2 (McGraw-Hill, New York 1954)

    Google Scholar 

  4. I. S. Gradshteyn, I. W. Ryzhik: Tables of Integrals, Series, and Products, 4th edn. (Academic Press, New York 1965)

    Google Scholar 

  5. E. Jahnke, F. Emde: Tables of Functions with Formulae and Curves, 4th edn. (Dover, New York 1945)

    MATH  Google Scholar 

  6. Y. L. Luke: The Special Functions and Their Approximations, Vol. 1, 2 (Academic Press, New York 1969)

    MATH  Google Scholar 

  7. Y. L. Luke: Mathematical Functions and Their Approximations (Academic Press, New York 1975)

    MATH  Google Scholar 

  8. W. Magnus, F. Oberhettinger, R. P. Soni: Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edn. (Springer, Berlin, Heidelberg 1966)

    MATH  Google Scholar 

  9. F. W. J. Olver: Asymptotics and Special Functions (Academic Press, New York 1974)  Reprinted A. K. Peters, Wellesley 1997

    Google Scholar 

  10. G. Szego: Orthogonal Polynomials, Vol. 23, 4th edn. (American Mathematical Society, Providence 1975)

    Google Scholar 

  11. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark: NIST Handbook of Mathematical Functions, in preparation

    Google Scholar 

  12. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark: NIST Digital Library of Mathematical Functions, in preparation (see http://dlmf.nist.gov/)

  13. A. R. Edmonds: Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton 1960) Sec. 2.5

    Google Scholar 

  14. L. Pauling, E. Bright Wilson: Introduction to Quantum Mechanics With Applications to Chemistry (McGraw-Hill, New York 1935) Sect. 21

    Google Scholar 

  15. G. W. F. Drake, R. N. Hill: J. Phys. B 26, 3159 (1993)

    ADS  Google Scholar 

  16. L. I. Schiff: Quantum Mechanics, 3rd edn. (McGraw-Hill, New York 1968) pp. 234-239

    Google Scholar 

  17. J. D. Talman: Special Functions, A Group Theoretic Approach (W. A. Benjamin, New York 1968) pp. 186-188

    MATH  Google Scholar 

  18. V. Fock: Z. Physik 98, 145 (1935) The relation of the Runge-Lenz vector to the “hidden” O(4) symmetry is discussed in Section 3.6.3

    Article  MATH  ADS  Google Scholar 

  19. M. Reed, B. Simon: Methods of Modern Mathematical Physics. I. Functional Analysis (Academic Press, New York, London 1972) pp. 263-264 Chapters VII The Spectral Theorem and VIII Unbounded Operators. See Theorem VIII.6

    Google Scholar 

  20. L. Hostler: J. Math. Phys. 5, 591, 1235 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  21. J. Schwinger: J. Math. Phys. 5, 1606 (1964)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. R. A. Swainson, G. W. F. Drake: J. Phys. A 24, 79, 95,1801 (1991)

    MATH  ADS  MathSciNet  Google Scholar 

  23. R. N. Hill, B. D. Huxtable: J. Math. Phys. 23, 2365 (1982)

    Article  MATH  ADS  Google Scholar 

  24. R. Courant, D. Hilbert: Methods of Mathematical Physics, Vol. 1 (Interscience, New York 1953)

    Google Scholar 

  25. D. J. Hylton: J. Math. Phys. 25, 1125 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  26. D. W. Lozier, F. W. J. Olver: Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics Vol. 48, ed. by W. Gautschi (American Mathematical Society, Providence 1994)

    Google Scholar 

  27. I. J. Thompson, A. R. Barnett: J. Comput. Phys. 64, 490 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. I. J. Thompson, A. R. Barnett: Comp. Phys. Commun. 36, 363 (1985)

    Article  MATH  ADS  Google Scholar 

  29. R. E. Meyer: SIAM Rev. 31, 435 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  30. H. Skovgarrd: Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions (Gjellerups, Copenhagen 1966)

    Google Scholar 

  31. N. Bleistein, R. A. Handelsman: Asymptotic Expansions of Integrals (Holt, Rinehart, & Winston, New York 1975) Reprinted Dover, New York 1986

    MATH  Google Scholar 

  32. R. Wong: Asymptotic Approximations of Integrals (Academic Press, San Diego 1989) Reprinted SIAM, Philadelphia 2001

    MATH  Google Scholar 

  33. R. C. Forrey, R. N. Hill: Ann. Phys. 226, 88-157 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. R. N. Hill: Phys. Rev. A 51, 4433 (1995)

    Article  ADS  Google Scholar 

  35. R. C. Forrey, R. N. Hill, R. D. Sharma: Phys. Rev. A 52, 2948 (1995)

    Article  ADS  Google Scholar 

  36. C. Krauthauser, R. N. Hill: Can. J. Phys. 80, 181 (2002)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 355 Laurel Avenue, 55102-2107, Saint Paul, MN, USA

    Robert Hill

Authors
  1. Robert Hill
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Hill .

Editor information

Editors and Affiliations

  1. Department of Physics, University of Windsor, 401 Sunset St., N9B 3P4, Windsor, ON, Canada

    Gordon Drake Prof.

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag

About this entry

Cite this entry

Hill, R. (2006). Hydrogenic Wave Functions. In: Drake, G. (eds) Springer Handbook of Atomic, Molecular, and Optical Physics. Springer Handbooks. Springer, New York, NY. https://doi.org/10.1007/978-0-387-26308-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation