Quantum Signatures of Solar System Dynamics

The role of celestial mechanics in development of modern quantum mechanics is well described in lecture notes by Born [1] . Surprisingly, usefulness of the atomic mechanics to problems of celestial mechanics has been recognized only very recently [2,3]. Closely related to these papers is the paper by Convay at al [4] where methods of optimal control and genetic algorithms were used for mission planning problems 1 . In this work we extend the emerging reverse trend. For this purpose it was nesessary to critically reanalyzed the logical steps leading from the old to new quantum mechanics in the light of available astronomical observational data. For the sake of uninterrupted reading, some not widely known facts from history of quantum mechanics are presented in nontraditional setting involving the latest results from mathematical and atomic physics. These facts are very helful for formulation of the problems to be solved in this paper. Thus, below, we discuss some historical background information first.

In 1923-24 academic year in Göttingen Max Born replaced planned two-semester lecture course in celestial mechanics by the course in atomic mechanics. Contrary to the standard superficial descriptions of “old” quantum mechanics which can be found at the beginning of any textbook on quantum mechanics, the achievements of “old” quantum mechanics go far beyond calculation of spectra of Hydrogen atom. In fact, the optical and X-ray spectra of almost all known at that time elements were found accounting even for the fine structure relativistic effects. The theory of quantum angular momenta was developed and used in the theory of polyatomic molecules. The effects of Zeemann and Stark were considered as well, etc. If one would make an itemized list of problems considered in “old” quantum theory and would compare it with that for “new” quantum theory, surprisingly, one would not be able to find an item which was not treated within the “old” formalism. With such an impressive list of acomplishments it is hard to understand why this formalism was abruptly abandoned in favour of “new” quantum mechanics in 1925. To explain this, we would like to bring some excerpts from the paper by Pauli and Born [5]. Being thoroughly familiar with works by Poincare ′ on celestial mechanics, they were trying to apply these methods to multielectron atoms. For this purpose they were using methods of theory of perturbations to account for electron-electron interactions. By doing so they obtained the same types of divergencies as were known already from calculations of planetary dynamics. By realizing the asymptotic nature of the obtained expressions, they decided that to keep just few terms in these expansions is the best way to proceed. By doing so a reasonably good agreement with experimentally known location of spectral lines was expected to be obtained. Such a state of affairs had caused frustration for Bohr who conceded that only those dynamical systems which admit a complete separation of variables are quantizable 2 . If such a separation is absent, according to Bohr’s current opinion, the system should not possess a discrete spectrum so that visible lines in spectra of elements other than Hydrogen should/must be much wider. On the theoretical side such an assumption calls for development of methods enabling to determine the widths of spectral lines and of distribution of the intensity within these widths. Such an intensity is expected to be connected with the underlying mechanical motion inside the atomic system.

Spectroscopical data for almost entire periodic system were readily available at the turn of the 20th century [6]. Bohr was well aware of these data and used them for his search for correct atomic model (along with Rutherford’s results of 1912 on scattering from the Hydrogen atom). In particular, he looked at the data for Helium in 1913 and published his findings in Nature [7]. For the sake of arguments which will follow, we found it helpful to reproduce some of the data from his Table 1 below. These data were compared with those for the Hydrogen for which he used the analogous table (Table 2) 3 . For some reason, the data in his Table 2 did not contain the error column. Since the wavelength λ in both cases was measurable, it was possible to evaluate the ratio K H /K He , where K= λ( 1

2 )•10 10 , which was found to be 4.0016. At the same time, Bohr’s own calculations gave for K the following value: K= c(M+m)h 3 2π 2 Z 2 e 2 Mm , with h being the Planck’s constant, Z and M being the charge and the mass of the nucleus, c being the speed of light and e and m are being the charge and the mass of the electron. By assuming M He = 4M H and Z He = 2Z H , one readily obtains for K H /K He the result: 4.00163. It is in good agreement with that obtained experimentally. In doing such calculations Bohr assumed that each electron in Helium can be treated as if it is a Hydrogen-like. This surely implies that the width o

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