Relativistic additions to the formalism of quantum mechanics. Part 3 of 5

RESEARCH ARTICLE

Kolisnyak Denis kolisnjakde@yandex.ru

http://kolisnjakde-rafqm.tilda.ws

III. Connection with nonrelativistic quantum mechanics.

Following the above model, let us find the differential relations to which it leads in case of low velocities. Taking into account relations (2.4) and (2.5) for energy and momentum, we obtain the following for the oscillation phase:

In nonrelativistic physics,

so:

We assume, as is customary in classical physics, that the time interval does not depend on the reference frame. For clarity, we carry out further calculations for one-dimensional motion. The transition to three-dimensional space is carried out the way described in the previous chapter.

We obtain the functions:

To describe the oscillation propagation in space, as before, let us parametrize these functions the following way:

While expanding momentum as a power series, we omit the l=vt value neglecting the point particle motion. In this case, the total momentum *p=h/λ* is determined only by value 1/λ*. Therefore, at x₁=const*, we observe the following oscillations:

The second function correlates to the point particle motion at the velocity of c²/v, or, in other words,

Relations (3.7) and (3.8) do no state that particle x₁ does not have a momentum. However, we do neglect the motion (Φ₁ oscillation propagation in space) at x₁ as compared to x₂.

Note the fundamental fact that the equation describing the oscillation propagation (2.14) will lead to defining the nonrelativistic momentum after substituting with function Φ₂ in the nonrelativistic form:

Therefore, using the found relationship between the energy and momentum eigenvalues:

for a quantum particle, we obtain a second-order equation for function Φ₂. To do this, we define the momentum and energy using the Φ₂ derivatives:

where

and the total free particle energy is defined by:

As a result, we have:

or

We substitute the time derivative with its value:

and obtain the required second-order equation:

It is easy to carry out similar calculations for generalization in case of three-dimensional space using the results from the previous chapter. Finally, we come to the equation:

If the oscillatory system is in a potential field, then based on the condition that the oscillation periods for points r₁ and r₂ must exactly coincide, it is an additional condition for a quantum particle in the nonrelativistic limit by virtue of relation (2.6). Therefore, according to (2.6), the choice of points r₁ and r₂ cannot be arbitrary. For example, if we consider the semiclassical problem on hydrogen energy levels for circular orbits, then within the quantum particles framework it is (2.6) that leads to quantization of the angular momentum and energy levels. In this case, r₁ and r₂ belong to the same set of points due to the fact that Φ₁ and Φ₂ describe the same observed particle.

Let E be the total point particle energy excluding mc². Then, in a potential field for a stationary state with kinetic energy Eₖ=E-Eₚ and subject to (2.6), equation (3.18) takes the form:

We parameterize Φ₁(t) by *λ* similarly to function Φ₂*, as done in the previous chapter. Then the function

describes oscillations Φ₁ and Φ₂ on scale *λ*. Function ξ₂ *also satisfies equation (3.19) on condition that in one state a relation equivalent to (2.6) is fulfilled:

It is well-known that the solution to equation (3.19) combined with the normalization of the wave function per unit as applied to the hydrogen atom leads to the correct energy levels [3].

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