# Relativistic additions to the formalism of quantum mechanics. Part 2 of 5

RESEARCH ARTICLE

Kolisnyak Denis

kolisnjakde@yandex.ru

**II. Kinematics of quantum particles.**

A quantum particle correlates to an oscillatory system with oscillation period *τ* and a momentum which depends on spatial parameters *λ** and *l*. As stated above, *l* is associated with the ordinary motion of a point particle. To answer the question about the physical meaning of *λ**, let us transfer to a nonrelativistic case.

In the energy and momentum formulae, let us take into account that *τ* depends on velocity:

then

Taking into account the connection between *τₒ* and the point particle mass, the energy and momentum take the well-known form:

Expanding equation (2.2) as a power series in *v/c* and omitting all terms with powers higher than second, we find:

Taking (1.4) into account, we obtain:

Thus, *E* and *p* have a standard velocity dependence typical of classical mechanics. Assuming that *v=const* and switching to the already used values *l* and *λ** we see that the transition to the classical case corresponds to neglecting the point particle motion and its contribution *l* to the energy and momentum values in formulae (1.3). The quantum particle momentum at *v<<c* is determined only by value *λ**, which is also a spatial parameter of the oscillatory system.

Let us introduce function *Φ₂* which correlates to one more part of the oscillatory system characterized by *λ** value. We define harmonic oscillations *Φ₁* and *Φ₂* as real functions, so that the sum of their squares gives some constant value:

This is due to the fact that only one particle cat actually be observed in an experiment at a time. For definiteness, let us define a specific form for *Φ₁* and *Φ₂* fitting (2.6):

As a result, in the general case, the oscillatory process corresponding to a quantum particle can be described by functions *Φ₁(t)* and *Φ₂(t)*, which must correspond to the second-order oscillatory equation. In general, it can be written as:

In a stationary state, or at constant energy, the oscillation propagation in space can be described when parametrizing the oscillations with parameters* l* and *λ**. Then in case of one-dimensional motion, we obtain:

For *Φ₂(t)* function:

where

Taking (1.3) into account, the functions become:

Also, for propagation oscillations *Φ₁(x₁(t))* and *Φ₂(x₂(t))*, the following differential relations are fulfilled:

This raises the question of how to orient oscillation propagations *Φ₁(x₁(t))* and *Φ₂(x₂(t))* relative to each other. Must the directions of the propagation match? At this point, we will assume that they must.

To generalize the discussion for the case of three-dimensional motion, consider the case of motion in three-dimensional space along the *x*-axis. At the initial moment of time *t=0*, let function *Φ₁* at point *O(0,0,0)* have an extremum. Based on the previous reasoning, we assume that the probability density determined by *Φ₂²* equals to zero. Assume that at this moment of time it is possible to find the particle in *O(0,0,0)*. Further oscillation propagation along the *x*-axis will change the situation according to (2.9), (2.10).

On the other hand, if we direct the velocity characteristic of *Φ₁(x₁(t))* to zero, then periodicity length* λ* *will tend to infinity; in other words, *Φ₂* will “straighten out”.

When moving in three-dimensional space along the *x*-axis, only one component of velocity vector, *vₓ*, is nonzero. This means that *Φ₂* “straightens out” for a given *x* along the line passing *O* straight across the *x*-axis.

In other words, in three-dimensional space, we can determine function *Φ₂(**r₂**(t))* as:

It means that in three-dimensional space for oscillation *Φ₁* along the *x*-axis, there are many parallel trajectories along which oscillation *Φ₂* can propagate, and (2.15) allows us to consider this. As stated above, the presence of an extremum at a point for functions *Φ₁(x₁(t))* or *Φ₂(**r₂**(t))* corresponds to the presence of a particle in it. In three-dimensional space, because of relation (2.15), a particle can be found not only at the set of points belonging to the *x*-axis, even if assuming that oscillation *Φ₁* propagates strictly along the *x*-axis. This suggests that the trajectories (as far as this term can be used even within this article) of the actually observed particles in the stationary state cannot be straight lines.

Finally, assuming that range of parallel trajectories *x₁* leads to range of parallel trajectories *x₂*, function *Φ₁* can be written as:

Then the system (2.13), (2.14) describing oscillations becomes:

Vectors ** v₁** and

**in this case are equal, and**

*v₂**|*

*v₁**|=|*

*v₂**|=v*.

Introducing the derivatives into system (2.17) we can verify that the system is equivalent to two relations of the type:

It is also worth noting that with introducing vectors ** k₁** and

**the following way:**

*k₂*for oscillations *Φ₁(**r₁**(t)) *and *Φ₂(**r₂**(t))* it can be written:

Then the eigenvalues of the 4-momentum component for *Φ₁(**r₁**(t)) *and *Φ₂(**r₂**(t))* equal to:

Here we have introduced the following notation:

Therefore, it is possible to associate the eigenvalue 4-vector of the momentum to *Φ₁(**r₁**(t)) *and *Φ₂(**r₂**(t))*, or, as is customary in quantum theory, the 4-dimensional wave vector. This means that each oscillation corresponds to an abstract wave, which is characterized by the same eigenvalues of energy and momentum.

The choice of functions *Φ₁(**r₁**(t)) *and *Φ₂(**r₂**(t))* (2.7) is obviously not the only possibility. It is also obvious that the functions can be swapped with the same result. Moreover, we can find a slightly different form for the functions described by equation system (2.17). From the point of view of the relationship between energy and momentum, we can use the same complex functions written in two different special scales determined by parameters $l$ and *λ**. Consider, for example, functions *ξ₁* and *ξ₂* defined as follows:

Functions

differ from *Φ₁* and *Φ₂* only in scale.

is parameterized by *λ**,

is parametrizedby *l*. If *Φ₁(t)* and *Φ₂(t)* are two oscillatory processes on different scales, then each of the functions *ξ₁* and *ξ₂* describe the presence of two oscillatory processes on the same scale. With this, it is obvious that functions *ξ₁* and *ξ₂* also correspond to system (2.17).

**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