Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the potentialx 2N and the correspondence principle

  • 48 Accesses

  • 11 Citations

Abstract

Eigenenergies and frequencies are obtained for a particle oscillating in the potential (1/2)k N × 2N, wherek is a constant,x is displacement, andN is a real number. These potentials include the harmonic oscillator (N = 1) and the square well (N = ∞). Then th eigenenergy has the formA N n λ(N), whereλ(N) = 2N/(N + 1), andA N is independent ofn. Application is made to the correspondence principle for the potentialsN > 1 and it is concluded the classical continuum is not obtained in Bohr's limitn → ∞. Complete correspondence is attained in Planck's limith → 0, so that for these configurations the limitsh → 0 andn → ∞ are not equivalent. A classical analysis of these potentials is included which reveals the relation log E (ν/ν N ) = (N − 1)/2N between frequencyv and energyE, where the constantν N is independent ofE.

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

References

  1. Bohr, N. (1914).Fysisk Tedsskrift,12, 97; (1920).Zeitschrift für Physik,2, 423.

  2. Goldstein, H. (1959).Classical Mechanics. Addison Wesley, Reading, Massachusetts.

  3. Jammer, M. (1966).The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York.

  4. Landau, L., and Lifshitz, L. (1958).Quantum Mechanics. Addison Wesley, Reading, Massachusetts.

  5. Liboff, R. L. (1975).Foundations of Physics,5, 271; see also, “On the Validity of the Bohr Correspondence Principle,” to be published in,Annals de la Fondation L. de Broglie, Paris.

  6. Planck, M. (1906).Vorlesungen über die Theorie der Wärmestrahlung. Barth, Leipzig.

  7. Schiff, L. I. (1968).Quantum Mechanics, 3rd ed. McGraw-Hill, New York.

  8. Sommerfeld, A. (1915).Münchener Berichte, 425.

  9. ter Haar, D. (1964).Selected Problems in Quantum Mechanics. Academic Press, New York.

  10. van der Waerden, B. (1968).Sources of Quantum Mechanics. Dover, New York.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liboff, R.L. On the potentialx 2N and the correspondence principle. Int J Theor Phys 18, 185–191 (1979). https://doi.org/10.1007/BF00670395

Download citation

Keywords

  • Field Theory
  • Real Number
  • Elementary Particle
  • Quantum Field Theory
  • Harmonic Oscillator