The Classical Limit of the Quantum Kepler Problem

A. Martín-Ruiz, J. Bernal, Alejandro Frank, Adrián Carbajal-Dominguez

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, México D.F., México.
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, México D.F., México;Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, México D.F., México.
Universidad Juárez Autónoma de Tabasco, División Académica de Ciencias Básicas, Cunduacán, México.

DOI: 10.4236/jmp.2013.46112                

Abstract

The classical limit of the quantum mechanical Kepler problem is derived by using a simple mathematical procedure recently proposed. The method is based both on Bohr’s correspondence principle and the local averages of the quantum probability distribution. We illustrate in a clear fashion the difference between Planck’s limit and Bohr’s correspondence principle. We discuss the confinement effect in macroscopic systems.

Keywords

Kepler Problem; Classical Limits

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. J. Makowski, European Journal of Physics, Vol. 27, 2006, pp. 1133-1139. doi:10.1088/0143-0807/27/5/012
[2]	L. E. Ballentine, “Quantum Mechanics: A Modern Development,” World Scientific, New York, 1998.
[3]	L. E. Ballentine, Y. M. Yang and J. P. Zibin, Physical Review A, Vol. 50, 1994, pp. 2854-2859. doi:10.1103/PhysRevA.50.2854
[4]	M. Berry, Physica Scripta, Vol. 40, 1989, pp. 335-336. doi:10.1088/0031-8949/40/3/013
[5]	L. S. Brown, American Journal of Physics, Vol. 40, 1972, pp. 371-376. doi:10.1119/1.1986554
[6]	L. S. Brown, American Journal of Physics, Vol. 41, 1973, pp. 525-530. doi:10.1119/1.1987282
[7]	D. Bhaumik, B. Dutta-Roy and G. Ghosh, Journal of Physics A: Mathematical and General, Vol. 19, 1986, pp. 1355-1364. doi:10.1088/0305-4470/19/8/017
[8]	S. Nandi and C. S. Shastry, Journal of Physics A: Mathematical and General, Vol. 22, 1989, pp. 1005-1016. doi:10.1088/0305-4470/22/8/016
[9]	G. Yoder, American Journal of Physics, Vol. 74, 2006, p. 404. doi:10.1119/1.2173280
[10]	R. W. Robinett, American Journal of Physics, Vol. 63, 1995, pp. 823-832. doi:10.1119/1.17807
[11]	E. G. P. Rowe, European Journal of Physics, Vol. 8, 1987, pp. 81-87. doi:10.1088/0143-0807/8/2/002
[12]	A. R. Edmonds, “Angular Momentum in Quantum Mechanics,” Princeton University Press, Princeton, 1974.
[13]	J. Bernal, A. Martn-Ruiz and J. Garca-Melgarejo, Journal of Modern Physics, Vol. 4, 2013, pp. 108-112. doi:10.4236/jmp.2013.41017
[14]	H. Goldstein, C. P. Poole and J. P. Safko, “Classical Mechanics,” Addison-Wesley, San Francisco, 2002.
[15]	R. Liboff, “Introductory Quantum Mechanics,” 4th Edition, Addison-Wesley, Boston, 2002.
[16]	H. M. Srivastava, H. A. Mavromatis and R. S. Alassar, Applied Mathematics Letters, Vol. 16, 2003, pp. 1131-1136. doi:10.1016/S0893-9659(03)90106-6
[17]	A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, “Integrals and Series, Vol. 3: More Special Functions,” Gordon and Breach Science Publishers, New York, 1989.
[18]	L. J. Slater, “Generalized Hypergeometric Functions,” Cambridge University Press, Cambridge, 2008.
[19]	M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables,” Dover Publications, New York, 1965.
[20]	H. Bacry and I.-M. Lavy-Leblond, Journal of Mathematical Physics, Vol. 9, 1968, pp. 1605-1615. doi:10.1063/1.1664490
[21]	W. H. Zurek, Reviews of Modern Physics, Vol. 75, 2003, pp. 715-775. doi:10.1103/RevModPhys.75.715
[22]	D. Sen and S. Sengupta, Indian Journal of Physics, Vol. 73, 1999, pp. 135-139.
[23]	A. Martín-Ruiz, J. Bernal and A. Carbajal-Dominguez, “Residual Effects of Quantum Transitions at Macroscopic Level,” in Press.