V.6 No 1

7

On excited state of orbital electron

2. Change of Kepler’s law when electron’s orbital motion in the external field

To enter the meaning of the issue, consider, how Keplerian law of areas changes for the hydrogen atom interacting with the electric field of general type, and whether this law changes in such interaction. We yet will not account the orbital motion of the atomic nucleus, we will confine ourselves to a simple model of electron orbiting around the nucleus at rest, as shown in Fig. 6.

Fig. 6. A simplified kinematic scheme of a hydrogen atom in the external time-variable electric field

The modelling system of differential equations describing the electron’s motion in this model is

where e is the electron’s charge, ₀ is the dielectric constant, m_e is the mass of electron, r is the radius-vector of the orbital electron in the frame of atom, _e and _E are the frequency and initial phase of the external time-variable electric field.

On one hand, the system (35) relates to the trivial modelling of processes in Bohr’s theory. Really, the trivial modelling system of equations is

[7, p. 20- 21]. We see that (35) and (36) differ only by the chosen reference frame - (35) is in Descartes and (36) in the polar system.

On the other hand, in (35), except the central field, the external force is also involved, and this distinction matters, as it immediately in the modelling equation accounts the excitation, while the conventional modelling (36) operated only with the stationary frame and estimated the excitation by the level of orbital electron energy change between the stationary states; as we will see below, this is not sufficient.

In absence of external field, both frames are identical and describe generally the elliptic trajectory of electron’s motion.

In presence of external field, we in general case may not fully identify (35) with linear oscillator, and we can show it mathematically.

To do so, multiply the first equation of (35) into y and the second into x, and subtract this first from the second; we will yield

Given

the left part of (37) will take the following form:

With it we can write (37) as follows:

To go on transforming (40), we have to pass to the polar coordinate system, as it trivially follows:

Substituting (41) into (40), yield

We see in the left part of (42) the derivative in time of the sectorial speed of electron. Actually, it is known that the sectorial speed of the orbiting body is determined by the vectorial product

where is the orbital speed of the body.

In its turn, in the plane case, general speed of the body in polar coordinates is

Substituting (44) into (43), yield

and this proves our statement.

Thus, given (43) and (44), (42) takes the sought form

In the absence of external field, i.e. when

it follows from (46) that

which is the mathematical record of the second Kepler law that says, in orbital motion the sectorial speed of the body is constant. While in presence of the external field, it follows from (46) that the second Kepler law is transformed, due to which the sectorial speed gains periodicity. When the body moved with the force (the radial angle varies in the first half of period), the sectorial speed grows, and when the body moved against the force, it decreases. With it the sectorial speed is maximal at the given angle equal to and minimal at = 0 . In its turn, the sectorial acceleration is maximal at = /2 and minimal (which means the maximal brake) at = 3/2 . Due to this, the ellipticity of orbit is generally distorted.

Now if we premise that in approximation of small amplitude of the external field the momentary phase that determines the electron’s location in the orbit is approximately determined by the periodic function of time

then in the right part of (46) we have the superposition of two oscillations,

It immediately follows from (49) that maximal transformations of electron’s trajectory will relate to the proportional frequencies of the external field and electron’s orbiting. Namely so, as we cited above, Bohr in description of excitation wrote: “Now suppose that the electron emits the monochromatic radiation with the frequency equal to a half of electron’s orbital frequency in its final orbit” [12, p. 87]. And namely so the initial frequency of electron’s orbiting was of no matter for Bohr: “If we assume the radiation to be monochromatic, there inevitably arises the second assumption concerning the radiation frequency - namely, that the number of electron’s turns in the beginning of radiation is zero” [12, p. 87]. From the view of (49), Bohr’s assumption says only that in this case not the frequency of electron’s motion in its stationary orbit is of importance but this frequency’s interrelation with the external excitation frequency. Just in this case we will see resonance phenomena in the system and the atom as a rotating dipole will radiate outwards most effectively. If the frequencies were not multiple, there, as in any dynamic system, oscillations will be non-stationary and the radiation will considerably decrease.