Relativistic additions to the formalism of quantum mechanics. Part 4 of 5

RESEARCH ARTICLE

Kolisnyak Denis kolisnjakde@yandex.ru

http://kolisnjakde-rafqm.tilda.ws

IV. Connection with relativistic quantum mechanics.

To analyze the results and compare the principle of two scales of the quantum particle model with the Dirac’s quantum theory, let us first return to the remark made in Chapter II on the propagation orientation of oscillations Φ₁ relative to Φ₂. We carry out all the basic calculations for one-dimensional space, which are easy to generalize for the case of three-dimensional motion.

Let us write out the main results of the previous reasoning assuming that the directions of propagation Φ₁ relative to Φ₂ may differ. We have the following equations describing the oscillation propagation:

As stated above, vectors v₁ and v₂ are equal in modulus: |v₁|=|v₂|=v since the oscillations describe one and the same observed particle.

Functions Φ₁ and Φ₂, which depend on time t, have the form:

For stationary states, oscillation propagation in space is described by the same functions Φ₁ and Φ₂, which have the spatial period lengths of l and *λ**:

where

Substituting functions Φ₁ and Φ₂ into equations (4.1) and (4.2) leads to two relations of relativistic link between energy and momentum (2.18). For a one-dimension case, it is obvious that after substituting equations (3.1) for energy and momentum into (2.18) we obtain l and *λ** definitions:

Moreover, for the one-dimensional motion there are two possible cases, which we will consider separately: v₁↑↑v₂ and v₁↓↑v₂.

Let us consider the v₁↑↑v₂ case. For further comparison with the Dirac’s theory it is necessary to combine equations (4.1) and (4.2) into one. Simple addition or subtraction of equations (4.1) and (4.2) lead to trivial equations. Therefore, we multiply (4.2) by

and subtract the obtained equation from (4.1) (in addition we multiply (4.1) by 1/Φ₂ and multiply (4.2) by 1/Φ₁):

Similarly, we perform equation addition and find:

or, if we switch to one-dimension notation taking into account that |v₁|=|v₂|=v:

We substitute functions Φ₁ and Φ₂ into (4.11) and (4.12) and, through the previously obtained relation (2.3), we define energy through mass in the second term on the left, therefore finding two relations:

The two latter equations between energy and momentum are equivalent to the relativistic scalar product of the 4-momentum and 4-velocity of a point particle in the sense that for the same momentum they give equivalent energy:

We transform (4.16) for one-dimensional motion:

Further, we perform elementary calculations and find the equivalence of (4.17) to (4.13) and (4.14).

With functions ξ₁ and ξ₂, equations (4.11) and (4.12) take the following form due to the same dependence of the functions on time:

Finally, let us not that it is the oscillatory system where functions ξ₁ and ξ₂ have the same periodicity period l or *λ* that leads to relations (4.13) and (4.14), even if functions ξ₁ and ξ₂ are identical. In this case, following the logic of the previous chapter, the question of transition to the nonrelativistic limit remains open, since in this case it is the simultaneous presence of two parameters that makes spatial parametrization of ξ₁ impossible due to the smallness of l *and the correct transition to the nonrelativistic description of the oscillatory system.

Consider the case of v₁↓↑v₂. To describe it, it is necessary to repeat literally all the logical steps described above, with the only difference that functions Φ₁ and Φ₂ describe oscillation propagation in opposite directions in one-dimensional space. Let Φ₁ describe the oscillation propagation along x-axis, just like in the previous case. Then function Φ₁ retains exactly the same form as before:

and satisfies the equation:

Φ₂ oscillations describe the oscillation propagation in the direction opposite to the x-axis, so the form of Φ₂ will differ from the previous case. We will use the prime mark for Φ₂ oscillation parameters for v₁↓↑v₂:

The momentum has the following components: *p₂’=(-p₂,0,0); the p₂=(p₂,0,0) momentum correlates to the previous case, so p₂’=-p₂*.

Due to the fact that the point particle energy with momentum p₂’ can be represented as:

where *v₂’=(-v₂,0,0) is the velocity of a point object correlating to momentum p₂’, therefore, for Φ₂’, using the v₂=(v₂,0,0) speed correlating to momentum p₂ *and taking into account that

it can be written:

Therefore, the oscillation function in the observed case differs only in sign relative to the previous configuration; that is, Φ₂’=-Φ₂. The analysis of (4.22)— (4.24) allows us to say that formally Φ₂’ describes the oscillation propagation differing from Φ₂ only in the phase in the direction of the x-axis.

To obtain an equation for Φ₂’, consider equation (4.2) written in such coordinates that the x-axis coincides with direction of r₂’:

Carrying out a time inversion which changes the direction of oscillation propagation to the opposite and corresponds to the change of coordinates of r₂’→ r₂, we find the equation for function (4.24):

Substituting Φ₂’ to (4.24) leads to the correct relation between the energy and momentum (4.23).

Repeating the steps described for the v₁↑↑v₂ case as applied to equations (4.21) and (4.26), we find two equations that lead to the relations between energy and momentum (4.13), (4.14):

Using the functions ξ₁ and ξ₂’=-ξ₂:

we find that they also satisfy equations (4.27) and (4.28). We simplify the system, and for ξ₁ and ξ₂’=-ξ₂ it can be written:

Therefore, we have four equations for describing propagating oscillations Φ₁, Φ₂, Φ₁’, Φ₂’ in two configurations:

For complex functions ξ₁, ξ₂, ξ₁’, ξ₂’ described above we obtain:

The first two equations in system (4.33) describe the configuration where the oscillations with spatial parameters l and *λ* propagate in the same direction, while the second pair describe the oscillations propagating in the opposite directions. In addition, both functions in the second equation pair are marked with a prime, but, as mentioned above, ξ₁’ is fully equal to ξ₁*.

For further system (4.33) simplification let us return to the fact that from the point of view of the relation between energy and momentum in an equation system, it does not matter if the pairs of functions have the same parametrizations (l, for example) or, as described above, with l and *λ* parametrization. The correctness criterion in this case is the possibility of transition to the nonrelativistic case. Let us substitute the ξ₁, ξ₂, ξ₁’, ξ₂’* function values to system (4.33):

Let us introduce a notation:

and define the velocity modulus through γ. Then in (4.34) there appear coefficients before each term: either

which, with a nonrelativistic transition (series expansion) allow zeroing the contribution of one of the functions in each pair of ξ₁, ξ₂. Let us use this consideration and the fact that the form of the functions describing the oscillations is known in this case; then system (4.34) can be reduced to the following form with the use of parametrization with l value only:

As stated in the beginning of the article, the *Ψ** wave function, if it is complex conjugate to

and taken on the x₁=vt set of points, leads to ξ₁ oscillations. In other words,

Then by introducing the notation customary to relativistic quantum mechanics:

using system (4.35) and standard energy and momentum operators

we find:

or for a wave function, which we write as a column

we find:

Using matrices αₒ, α₁ of the following form:

we write the equation system in its final form:

Equation (4.39) is nothing else than Dirac equation written for the one-dimensional motion case. Herewith, matrices αₒ, α₁ are the Dirac gamma matrices satisfying the permutation relations of

is the Kronecker delta.

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