V.4 No 1

41

On orbital stability of oscillators

2. Standard statement of problem of electron moving in the central field of atomic nucleus, and its analysis

According to the statement of problem, to simplify the model of electron's motion in the central field of atom, Niels Bohr in [2] has introduced a number of quite just, in the view of mathematical physics, limitations:

First, "we will suppose that electron in the beginning of its interaction with the nucleus is far from the nucleus, and its velocity is not considerable as to the nucleus" [2,p. 87].

Second, "suppose further that after meeting the nucleus, the electron goes to the stationary orbit around the nucleus" [2, p. 87].

Third, "suppose the described orbit to be circular. This admission will cause no changes for systems containing only one electron" [2, p. 87].

Finally, fourth, "for the sake of simplicity, suppose the electron mass negligibly small in comparison with the nucleus mass, and the electrons velocity small as to the light velocity" [2, p. 86].

This last supposition has been introduced because of more general statement: "the problem of motion of system of two particles is equivalent to that of motion of one particle distanced to from the stationary centre and having the mass expressed through the masses of both particles as follows:

(5)

[1, p. 178]. "It would be justified to think the nucleus stationary only in case if we could think the nuclear mass infinitely large as to the electron mass. Whilst the mass of hydrogen nucleus related to the mass of electron as

(6)

and with highest precision of today spectroscopic measurements, it would be inadmissible to neglect that the nucleus mass is finite" [1, p. 338].

In the existing theories this circumstance is usually made more precise by way of introduction the corresponding correction in finding the Ridberg constant. "To solve precisely the problem of two bodies, we have in the formula

(7)

(where E_n is the quantum values of the energy for sequential row of integer values of n, m is the mass of electron, e is the charge of electron, Z is the quantity of electrons in atom, and h is the Planck constant - authors) to introduce instead the electron mass the reduced mass of electron and nucleus:

(8)

where M_Z  is the mass of nucleus with the atomic number Z. The formula

(where R is the Ridberg constant, and ? is the velocity of light in the void space - authors) will take the following form:

(9)

We would note, in the quantum-mechanical approach, the motion of nucleus is also neglected. Usually the statement of problem of hydrogen-like atom is as follows.

"Now consider, how the Schroedinger equation is applied to solve the problem of hydrogen-like atom. The potential energy of electron in the Coulombian field of nucleus will be in this case

(10)

Supposing the mass of nucleus infinitely large as to the mass of electron, we reduce the problem to the electron's motion around the stationary centre, i.e. to the problem of one body" [1, p. 547].

At the same time, the neglect of finite relation of proton and electron masses affects not only the mechanical model of system. We should note that the main distinction of atomic model from usual mechanical system is the presence of potential fields causing the interaction of electron and proton. With it even outwardly insufficient motion of field source is basically able not only to shift the mass centre of the system, but to change the conditions of charges interaction, and it is impossible to account this circumstance, introducing the correction into the final solution. Given this, we will concentrate our attention on studying the dynamic field of nucleus of hydrogen-like atom, in whose field the electron orbits.