where and are the Bessel functions of the first kind. We point out that we can calulate higher order approximations, but for our purposes we consider here only the first one.

The inverse Fourier transform of Equation (14) which can be found in Ref.[19] gives

(15)

which corresponds exactly with the classical radial probability density. The higher order corrections involve a series depending on successive powers of.

4. Discussion

When a theory succeeds in describing physical reality in a better way then its predecessor, as is the case of relativity compared with Newton’s theory, and both share an essential physical framework, a purely mathematical reduction of the more general theory to the restricted one is generally possible. Even in the case of the special theory of relativity, Bacry and Levy-Leblond [20] have shown that the low velocity limit of Lorentz transformations for space-like intervals display nonGalilean features, as the time-ordering of two such independent events may be reversed for some observers. The quantum-classical relation is more subtle, given that the conceptual frameworks of these theories are fundamentally different. The question that naturally arises is the way in which quantum theory reduces to classical theory when applied to a macroscopic system. Zurek, for example, studies the role that the enviroment has on producing the effect of decoherence [21].

In the classical regime, the probability distribution becomes entirely confined between two turning points and as is kept constant, momentum is also confined within a finite interval. Such a state is no longer describable by a wave function [22]. The quantum-mechanical case is radically different. On the one hand, the probability distribution is distributed in all space, so the probability that the particle is in the volume is given by

(16)

On the other hand, it is impossible to define quantum turning points (as in the classical sense) because of Heisenberg’s theorem works. These facts make these theories seem conceptually incompatible.

It is well known, however, that the probability distributions, and, approach each other in a locally averaged sense, when some appropriate quantum numbers become large [9-11]. Although this result seems simple, there is no simple mathematical procedure to prove this assertion. In a previous work [13], we introduced a mathematical procedure to prove that the quantum probability distribution leads to the classical one when applied to the high energy regime. Our method is based on both the Bohr’s correspondence principle and the local averages of the quantum probability distribution. Note that we do not need to consider the limit where. There exist higher order corrections which can be expressed as powers of. The classical result for the probability distribution is recovered as the -independent zeroth order term. In this approach Heisenberg’s theorem applies even at the macroscopic level. This is philosophically more satisfactory than actually using the limit and provides corrections that are associated in principle with the classical-quantum borderline.

As can be seen from Figure 1 and other cases like the harmonic oscillator, as the principal quantum number is increased, becomes spatially confined in a couple of points, then becomes a rapidly oscillatory function inside this region while the outside is strongly suppressed. We therefore observe macroscopically a motion bounded by this couple of points, identified as the classical turning points and the probability of finding the particle outside this region, i.e. Equation (16), tends to zero as the energy increases, so that it becomes forbidden in this regime. When we use our procedure, the oscillatory behavior of the quantum distribution is averaged out and becomes a smooth function in the macroscopically accessible region. The function cancels outside this region and the turning points look like infinite walls, akin to the macroscopic behavior.

5. Conclusions

Since the birth of quantum mechanics, prevails the belief that is the general theory, and that classical mechanics must be deducible from it; however there is not a satisfactory demonstration of this belief. The answer to that question took the form of the heuristic principle, known as the correspondence principle. Although the importance of correspondence principle is largely undisputed, there is far less agreement concerning how it should be defined. The approaches discussed in almost all textbooks (as WKB approximation and Ehrenfest’s theorem) are not generally valid. No doubt that the classical limit is not a simple problem.

In a previous paper we introduced a simple procedure to connect the classical and quantum probability distributions for the harmonic oscillator case and we argue its general validity. We now report analytical results for the Kepler problem. It is noteworthy that these results cannot be achieved by any other procedure. The main result of this paper is the emergence of semi-classical Bohr’s circular orbits from purely quantum mechanical data.

We consider our approach demonstrates that quantum mechanics is applicable in every scale of nature, and that the macroscopic world is a consequence of its asymptotic behavior in the high energy regime. Even though our approach gives the correct classical results for periodic quantum systems, it is far from the general solution to the classical limit problem. There are still related open problems needed for a general mathematical formulation of the classical limit problem, as the study of the unbound systems. We are currently exploring residual effects of quantum transitions at macrocopic level [23].

REFERENCES

- A. J. Makowski, European Journal of Physics, Vol. 27, 2006, pp. 1133-1139. doi:10.1088/0143-0807/27/5/012
- L. E. Ballentine, “Quantum Mechanics: A Modern Development,” World Scientific, New York, 1998.
- 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
- M. Berry, Physica Scripta, Vol. 40, 1989, pp. 335-336. doi:10.1088/0031-8949/40/3/013
- L. S. Brown, American Journal of Physics, Vol. 40, 1972, pp. 371-376. doi:10.1119/1.1986554
- L. S. Brown, American Journal of Physics, Vol. 41, 1973, pp. 525-530. doi:10.1119/1.1987282
- 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
- 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
- G. Yoder, American Journal of Physics, Vol. 74, 2006, p. 404. doi:10.1119/1.2173280
- R. W. Robinett, American Journal of Physics, Vol. 63, 1995, pp. 823-832. doi:10.1119/1.17807
- E. G. P. Rowe, European Journal of Physics, Vol. 8, 1987, pp. 81-87. doi:10.1088/0143-0807/8/2/002
- A. R. Edmonds, “Angular Momentum in Quantum Mechanics,” Princeton University Press, Princeton, 1974.
- 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
- H. Goldstein, C. P. Poole and J. P. Safko, “Classical Mechanics,” Addison-Wesley, San Francisco, 2002.
- R. Liboff, “Introductory Quantum Mechanics,” 4th Edition, Addison-Wesley, Boston, 2002.
- 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
- 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.
- L. J. Slater, “Generalized Hypergeometric Functions,” Cambridge University Press, Cambridge, 2008.
- M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables,” Dover Publications, New York, 1965.
- H. Bacry and I.-M. Lavy-Leblond, Journal of Mathematical Physics, Vol. 9, 1968, pp. 1605-1615. doi:10.1063/1.1664490
- W. H. Zurek, Reviews of Modern Physics, Vol. 75, 2003, pp. 715-775. doi:10.1103/RevModPhys.75.715
- D. Sen and S. Sengupta, Indian Journal of Physics, Vol. 73, 1999, pp. 135-139.
- A. Martín-Ruiz, J. Bernal and A. Carbajal-Dominguez, “Residual Effects of Quantum Transitions at Macroscopic Level,” in Press.