Foundations of Physics

, Volume 5, Issue 2, pp 271–293 | Cite as

Bohr correspondence principle for large quantum numbers

  • Richard L. Liboff


Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI−1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies.


Emission Spectrum Quantum Number Discrete Spectrum Periodic System Classical Ensemble 
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  1. 1.
    M. Jammer,The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), Section 3.2.Google Scholar
  2. 2.
    B. L. Van der Waerden,Sources of Quantum Mechanics (Dover, New York, 1968).Google Scholar
  3. 3.
    S. Tomonaga,Quantum Mechanics (North-Holland, Amsterdam, 1962), Vol. 1, Chapter 3.Google Scholar
  4. 4.
    R. Becker and F. Sauter,Quantum Theory of Atoms and Radiation (Blaisdell, New York, 1964), Vol. II.Google Scholar
  5. 5.
    R. Eisberg,Fundamentals of Modern Physics (Wiley, New York, 1961), Chapter 8.Google Scholar
  6. 6.
    D. ter Haar,Elements of Statistical Mechanics (Rinehart, New York, 1954).Google Scholar
  7. 7.
    L. I. Shiff,Quantum Mechanics (McGraw-Hill, New York, 1969), 3rd ed.Google Scholar
  8. 8.
    G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers (Oxford Univ. Press, London, 1960), 4th ed.Google Scholar
  9. 9.
    H. Goldstein,Classical Mechanics (Addison-Wesley, Reading, Massachusetts, 1959).Google Scholar
  10. 10.
    G. Birkhoff and G. Rota,Ordinary Differential Equations (Ginn, Boston, 1962), Chapter 10.Google Scholar
  11. 11.
    G. N. Watson,A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, London, 1966), 2nd ed.Google Scholar
  12. 12.
    T. Boyer,J. Math. Phys. 10, 1729 (1969).Google Scholar
  13. 13.
    C. W. Jones and F. W. Olver, inRoyal Society Mathematical Tables (Cambridge Univ. Press, 1960), Vol. VII, Chapter 1.Google Scholar
  14. 14.
    F. W. Olver,Phil. Trans. R. Soc. 247A, 307, 328 (1954).Google Scholar
  15. 15.
    R. L. Liboff,Introduction to the Theory of Kinetic Equations (Wiley, New York, 1969), Section 5.5.Google Scholar
  16. 16.
    A. d'Abro,The Rise of the New Physics (Dover, New York, 1939).Google Scholar
  17. 17.
    J. D. Jackson,Classical Electrodynamics (Wiley, New York, 1962).Google Scholar
  18. 18.
    L. Landau and L. Lifshitz,Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1958), Section 79.Google Scholar
  19. 19.
    C. S. Chang and P. Stehle,Phys. Rev. A 8, 318 (1973).Google Scholar

Copyright information

© Plenum Publishing Corporation 1975

Authors and Affiliations

  • Richard L. Liboff
    • 1
  1. 1.Schools of Electrical Engineering and Applied PhysicsCornell UniversityIthaca

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