Relativistic additions to the formalism of quantum mechanics. Part 5 of 5

RESEARCH ARTICLE

Kolisnyak Denis kolisnjakde@yandex.ru

http://kolisnjakde-rafqm.tilda.ws

V. Conclusion.

In this work, a model has been proposed, according to which a mass particle is an oscillatory system characterized by at least two coordinate sets

and

The observed point particle is one of the states of this oscillatory system. If a particle is a combination of two oscillation processes, then its various parts must have the same time; therefore, oscillation periods Φ₁(t) and Φ₂(t) coincide and are equal to τ. In other words, a quantum particle is the process of transition of a particle (or it’s energy) from

to a particle located in

and back in a time equal to τ. The equations, which describe the oscillatory system exactly correspond to the Schrodinger and Dirac equations in the nonrelativistic and relativistic cases, respectively.

The coordinate sets

and

themselves, as shown, can be the same regardless of whether the oscillations propagate in the same or opposite directions. In this case, comparing system (4.35) to Dirac equation written for the one-dimensional motion of a free particle, we come to the conclusion that the first pair of equations (4.35) describe the motion when the spin is oriented along the motion direction (along the $x$-axis), while the second pair describe the state where the particle spin is directed opposite the x-axis. However, the first pair (4.35) correlates to the case when the Φ₁(t) and Φ₂(t) oscillations propagate in one direction, along the x-axis, while the second pair represents the case when Φ₁(t) propagates along the x-axis and Φ₂(t) propagates against it. Therefore, we obtain a simple interpretation of two states of the same particle with the oppositely directed spin.

It is worth noting that in the transition from equation system (4.33) to (4.34), only one spatial parameter l retains for function pairs ξ₁, ξ₂, though initially it was assumed that there were two parameters: l and *λ**. Such transition does not affect the fundamental relationship between energy and momentum but influences, for example, the problem solution on energy levels in a hydrogen atom. The system with one parameter leads to the situation where two spin states lead to the same electron energy levels in a hydrogen atom. The presence of two parameters leads to the situation where two states with different spin directions give different levels of electron energy.

Let us demonstrate this consideration in more detail. For this, we will use the relativistic generalizations of Bohr’s model for the hydrogen atom, a method which is not the most accurate but is the most suitable for the purpose of greater clarity.

The relativistic equation of motion in electromagnetic fields has the form:

where

is the electromagnetic field tensor and

is the 4-velocity of an electron. Introducing centripetal acceleration and potential energy

we find the condition of motion on a circular orbit and motion energy:

where

is the modulus of the spatial part of the momentum 4-vector

or

The condition for the presence of one particle on a circular orbit is reduced to condition (2.6). In other words, if the oscillations propagate in the opposite directions, we obtain:

here, L=2πr is the circumference of a correct circular orbit.

If the oscillations propagate in the same direction, we obtain the second condition for quantizing circular orbits:

For the second condition, we can find the electron energy using the relation between the momentum and periodicity parameters l and *λ**:

where a notation is introduced for the constant of fine-structure constant:

In this case, the conditions of quantizing the orbits match the nonrelativistic condition of quantizing the electron’s angular moment.

For the electron energy excluding mc², we expand the function as a power series in α, and omit terms with powers higher than fourth, and find:

The condition of quantizing orbits for the first configuration, taking (1.3) into account, takes the following form:

This relation leads to a cubic equation for the propagation velocity modulus of oscillations correlating to electron v:

Finding real solutions to (5.10) and substituting them into energy (5.4), we carry out the expansion similar to the previous case. Therefore, we obtain the electron energy for the first configuration, which is the oscillations propagating in the opposite directions:

Energy value (5.11) at n₁=1 is omitted because the energy is positive in this case, which means that the electron is free. Therefore, the first pair of solutions corresponds to n₁, n₂=2:

The frequency difference for the two indicated energy levels is 4.4 ⋅ 10⁴ MHz, which is extremely large and does not correspond to the existing experimental data. However, it is worth noting that the method used in calculations is estimating; for these purposes, obtaining and solving wave equation (4.32), which is generalized for three-dimensional motion cases, is required. At the same time, such an estimation leads to the fact that when using the proposed model for calculating the levels of a hydrogen atom of the

type, the indicated levels have different energies. This fact comes from Lamb’s and Retherford’s experiments but is not presented in the solution to Dirac equation [4].

References

[1] A. Einstein, B. Podolsky and N. Rosen, Physical Review 47, 777 (1935).

[2] L. de Broglie, Foundations of Physics 1, 6,7 (1970).

[3] L. D. Landau, E. M. Lifshitz, Quantum mechanics non-relativistic theory, Vol.3, (Pergamon Press Ltd., Headington Hill Hall, Oxford, England, 1965), pp.118.

[4] V. B. Berestetskii, E. M. Lifshitz, L.P. Pitaevskii Quantum electrodynamics, Vol.4, (Pergamon Press Ltd., Headington Hill Hall, Oxford, England, 1982), pp.127.

© Kolisnyak D.E., 2022