Abstract
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.
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References
M. Jammer,The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), Section 3.2.
B. L. Van der Waerden,Sources of Quantum Mechanics (Dover, New York, 1968).
S. Tomonaga,Quantum Mechanics (North-Holland, Amsterdam, 1962), Vol. 1, Chapter 3.
R. Becker and F. Sauter,Quantum Theory of Atoms and Radiation (Blaisdell, New York, 1964), Vol. II.
R. Eisberg,Fundamentals of Modern Physics (Wiley, New York, 1961), Chapter 8.
D. ter Haar,Elements of Statistical Mechanics (Rinehart, New York, 1954).
L. I. Shiff,Quantum Mechanics (McGraw-Hill, New York, 1969), 3rd ed.
G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers (Oxford Univ. Press, London, 1960), 4th ed.
H. Goldstein,Classical Mechanics (Addison-Wesley, Reading, Massachusetts, 1959).
G. Birkhoff and G. Rota,Ordinary Differential Equations (Ginn, Boston, 1962), Chapter 10.
G. N. Watson,A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, London, 1966), 2nd ed.
T. Boyer,J. Math. Phys. 10, 1729 (1969).
C. W. Jones and F. W. Olver, inRoyal Society Mathematical Tables (Cambridge Univ. Press, 1960), Vol. VII, Chapter 1.
F. W. Olver,Phil. Trans. R. Soc. 247A, 307, 328 (1954).
R. L. Liboff,Introduction to the Theory of Kinetic Equations (Wiley, New York, 1969), Section 5.5.
A. d'Abro,The Rise of the New Physics (Dover, New York, 1939).
J. D. Jackson,Classical Electrodynamics (Wiley, New York, 1962).
L. Landau and L. Lifshitz,Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1958), Section 79.
C. S. Chang and P. Stehle,Phys. Rev. A 8, 318 (1973).
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This research was supported by the Physics Branch of the Office of Naval Research under Contract N00014-67-A-0077-0015.
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Liboff, R.L. Bohr correspondence principle for large quantum numbers. Found Phys 5, 271–293 (1975). https://doi.org/10.1007/BF00717443
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DOI: https://doi.org/10.1007/BF00717443