Relativistic additions to the formalism of quantum mechanics. Part 2 of 5
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:
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:
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 v₂ in this case are equal, and |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 k₂ the following way:
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:
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).
 A. Einstein, B. Podolsky and N. Rosen, Physical Review 47, 777 (1935).
 L. de Broglie, Foundations of Physics 1, 6,7 (1970).
 L. D. Landau, E. M. Lifshitz, Quantum mechanics non-relativistic theory, Vol.3, (Pergamon Press Ltd., Headington Hill Hall, Oxford, England, 1965), pp.118.
 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