## Abstract

A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.

### Keywords

- quantum mechanics
- relativistic invariant equations

## 1. Introduction

In 1913, Bohr, based on the Balmer empirical formulas, constructed a model of atom based on the quantization of the orbital momentum [1], which was subsequently supplemented by the more general Sommerfeld quantization rules. In those years, naturally, the presence of a spin or an intrinsic magnetic moment of the particle or, especially, spin-orbit interaction, or interaction with the nuclear spin, was not supposed.

In 1916, Sommerfeld, within the framework of relativistic approaches, derived a formula for the energy levels of a hydrogen-like atom, without taking into account the spin [2]. Sommerfeld proceeded from the model of the Bohr atom and used the relativistic relation between the momentum

where

In an external field with a four-dimensional potential (

In the case of the Coulomb potential

where

where

where the principal quantum number * made a fortunate mistake*’ [4] and the derived formula was presented in the following form

The formula (6) perfectly described all the peculiarities of the structure of the spectrum of hydrogen and other similar atoms with the limiting for those years accuracy of measurements, and there was no doubt about the correctness of the formula itself. Therefore, the Sommerfeld formula was perceived as empirical, and instead of the quantum number l, a ‘* mysterious*’ internal quantum number with half-integer values

where

Formula (7) also indicated a strange limitation of value the charge of a nucleus with the atomic number

In 1925–1926, Schrödinger worked on the derivation of the equation for the wave function of a particle describing the De Broglie waves [6]. The derivation of the equation also was based on the relativistic relation (1) between the momentum

Like Sommerfeld, Schrödinger used the following representation for a particle in an external field

In the case of stationary states of a charged particle in the field of the Coulomb potential for a hydrogen atom it was necessary to solve the equation

As can be seen, the quadratic expression of potential energy

Next, the wave vector

and when considering the problem of the passage of a particle with energy

Another difficulty is that, as the solution of the particle problem in a potential well shows, at a sufficient depth, a particle with a wavelength

Also, the solution of the problem of a hydrogen-like atom is limited by the value of the ordinal number of the atomic nucleus * particle fall on the center*” [8].

In order to get rid of the quadratic term or reverse its sign, in recent years it has been proposed to represent potential energy in the Klein-Gordon and Dirac equations as the difference of squares from the expressions of scalar and vector potentials (S-wave equation) [9, 10, 11]. Such a mathematical formalism corrects the situation, but from a physical point of view such representations are in no way justified, and the fields corresponding to such pseudo-potentials do not exist in nature.

Things are even worse with the presence of a quadratic term of the vector field, because of the sign of which we obtain non-existent states in nature and solutions that contradict experience.

According to the solutions of the equations of quantum mechanics and Hamilton-Jacoby, it turns out that a charged particle in a magnetic field, in addition to rotating in a circle, also has radial vibrations—Landau levels [12] (even in the case of zero orbital momentum).

Over these 90 years, especially in very accurate cyclotron resonance experiments, none has detected the electron radial vibrations and the Landau levels.

Solving this equation, Schrödinger, like Sommerfeld, received the formula (5), which described the structure of the hydrogen spectrum not exactly. Moreover, from the solution of the problem for a particle in a potential well, it turns out that a particle with a wavelength

In 1925 Schrödinger sent this work to the editors of ‘Annalen der Physik’ [13], but then took the manuscript, refused the relativistic approaches and in 1926 built a wave equation based on the classical Hamiltonian expression, the Schrödinger equation [14].

Equation described the spectrum of the hydrogen atom only qualitatively, however, it did not have any unreasonable restrictions or singular solutions in the form of the Sommerfeld-Dirac formula. Klein [15], Fock [16] and Gordon [17] published the relativistic equation based on the wave equation for a particle without spin in 1926; it is called the Klein-Fock-Gordon equation.

With the discovery of the spin, the situation changed drastically, and in 1926 Heisenberg and Jordan [18] showed that, within the Pauli description of the spin of an electron, half the energy of the spin-orbit interaction is equal to a term with a power of α^{4} in the Taylor series expansion of the Sommerfeld formula * equation reference goes here*.

Why exactly the half, Thomas tried to explain this in 1927 by the presence of a relativistic precession of an electron in the reference frame of motion along the orbit [19]. The energy of the Thomas precession is exactly equal to half the value of the energy of the spin-orbit interaction with the inverse (positive) sign, which should be added to the energy of the spin-orbit interaction. However, the incorrect assumption that the Thomas precession frequency is identical in both frames of reference and the absence of a common and correct derivation for non-inertial (rotating) frames of reference raised doubts about the correctness of such approaches. The reason for the appearance of half the energy of the spin-orbit interaction in the Sommerfeld formula is still under investigation and is one of the unresolved problems in modern physics.

On the other hand, both in the derivation of the Sommerfeld formula and at the solution of the Klein-Fock-Gordon equation for the hydrogen atom problem [20], neither the spin nor the spin-orbit interaction energy was taken into account initially. Therefore, the obtained fine splitting can in no way be owing to the spin-orbit interaction. This is a relativistic but purely mechanical effect, when the mass (inertia) of a particle is already depends on the velocity of motion along the orbit (of the angular momentum), because of which the radial motion of the electron changes, and vice versa. Just this dependence, which results in the splitting of the energy levels of the electron, and to the impossibility of introducing only one, the principal quantum number. Nevertheless, even with this assumption, the order of splitting of the levels according to formula (8) contradicts to the logic; it turns out to be that the greater the orbital angular momentum, the lesser the energy of the split level.

The matrix representation of the second-order wave Eq. (9) by a system of equations of the first order is the Dirac construction of the relativistic electron equation [21] (the Dirac matrices are the particular representation of the Clifford-Lipschitz numbers [22]). In the standard representation the Dirac equation for a free particle has the form [23].

where

are the Pauli matrices (the unit matrix in the formulas is omitted).

For a particle in an external field, Eq. (16) is usually written in the form

where for an invariant representation in the case of a free particle, the equations are composed for the difference between the generalized momentum and the momentum of the field.

In the case of the potential energy of an immobile charge in a Coulomb field, we obtain the Sommerfeld-Dirac formula as a result of an exact solution of this particular equation. There, again, although for a system with spin

More accurate measurements of Lamb in 1947 and subsequent improvements in the spectrum of the hydrogen atom revealed that, in addition to the lines with the maximum j, all the others are also split and somewhat displaced (the Lamb shift). To harmonize the results of the theory with more accurate experimental data on the spectrum of the hydrogen atom, one had to propose other solutions and approaches than were laid down by the derivation of the Dirac equation.

The new theoretical approaches had yield nothing and only supplemented the theory with the illogical and non-physical proposals to overcome the emerging singularity of solutions: the renormalization, the finite difference of infinities with the desired value of the difference, and so on. The accounting for the size of the nucleus corrected only the

The results of solution of the problem for a particle in a potential well both in the case of the Klein-Fock-Gordon equation and of the Dirac equation contradict to the basic principle of quantum mechanics, to the uncertainty principle. From the solutions, it turns out to be that a particle can be in a bound state in a well with any dimensions, in particular, with the size much smaller than the wavelength of the particle itself,

Despite Dirac himself proposed a system of linear first-degree relativistic equations in the matrix representation that described the system with spin

The reason for the failure of these theories is quite simple—it is in the ignoring of the dependence of the interaction energy with the field on the velocity of the particle. The generalized momentum of the system, the particle plus the external field, is the sum of the relativistic expression for the mechanical momentum of the particle and the field momentum in the case of interaction with the immobile particle

which is not an invariant representation of the particle velocity. To construct some invariant from such a representation, an ‘invariant’ relation was used in all cases in the form of a difference between the generalized momentum of the system and the field momentum in the case of interaction with the immobile particle

Obviously, the permutation of the components of the generalized momentum for the construction of the invariant does not solve the posed problem. The statement that the expression (20) is the mechanical momentum of a particle and therefore is an invariant is unproven and it is necessary to apprehend the formula (20) as an empirical. Therefore, at high velocities or strong interactions, an unaccounted dependence of the energy of particle interaction with the field on the velocity of the particle motion, which results to the erroneous results or the impossibility of calculations.

In [26], an invariant representation of the generalized momentum of the system was suggested, where the dependence of the interaction energy of the particle with the field on the velocity was taken into account:

which is the four-dimensional representation of the generalized momentum of the system based on the expression for the generalized momentum of an immobile particle in a state of rest

whose invariant is always equal to the expression (19) regardless of the state of the system.

The application of variational principles to construct the relativistic and quantum theory was based on the principles of construction the mechanics with the help of the Lagrangian of the system [27], which originally was not intended for relativistic approaches. The Lagrangian construction is parametric with the one time variable τ = ct, singled out from the variables of the four-dimensional space (the rest are represented by the dependence on this variable τ) and contains the total differential with respect to this variable, the velocity of the particle. Such a construction is unacceptable because of the impossibility to apply the principle of invariance of the representation of variables and the covariant representation of the action of the system.

In [28], to construct the relativistic theory on the basis of variational principles, the canonical (non- parametric) solutions of the variational problem for canonically defined integral functionals have been considered and the canonical solutions of the variational problems of mechanics in the Minkowski spaces are written. Because of unifying the variational principles of least action, flow, and hyperflow, the canonically invariant equations for the generalized momentum are obtained. From these equations, the expressions for the action function and the wave function are obtained as the general solution of the unified variational problem of mechanics.

Below, we present the generalized invariance principle and the corresponding representation of the generalized momentum of the system, the equations of relativistic and quantum mechanics [29], give the solutions of the problems of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), the problems of charged particle in a magnetic field, and also the problems of a hydrogen atom are given.

## 2. Principle of invariance

### 2.1 Generalization of the principle of invariance

The principle of invariance of the representation of a generalized pulse is applicable also in the case of motion of a particle with the velocity

The four-dimensional momentum of a particle

This is the property of invariance of the representation of the four-dimensional momentum ** P** in terms of the velocity of the particle

If to consider the representation of the four-dimensional momentum of an immobile particle with a mass

This is a property of invariance of the representation of the four-dimensional momentum

For an invariant of the system

At

Thus, the generalized momentum of a particle has an invariant representation on the particle velocity

If a charged particle is in an external electromagnetic field with potentials

The fact that the interaction of a charged particle with a field depends on the speed of motion is evidently represented in the formula for the Liénard-Wiechert potential [8].

More clearly, this can be demonstrated by an example of the Doppler effect for two atoms in the field of a resonant radiation, when one of the atoms is at rest and the other moves with the velocity

The atom, which is at rest, absorbs a photon, and the moving one does not absorb or interacts weakly with the field, because of the dependence of the interaction on the velocity of the atom. It is also known that the acting field for an atom moving with the velocity v corresponds to the interaction with the field moving with the velocity

### 2.2 Invariant representation of the generalized momentum

Thus, for a moving charge, the effective values of the potentials

If one represents the generalized momentum of the particle in the form

where

The expression (30) can be represented in the form

or

This transformation can be presents in matrices form

where a Lorentz transformation have a form

The matrices of the invariant representation of a four-dimensional vector, which preserve the vector module in four-dimensional space, form the Poincare group (inhomogeneous Lorentz group). In addition to displacements and rotations, the group contains space-time reflection representations

For the module

which is the four-dimensional representation of the generalized momentum of the system on the basis of the expression of the generalized momentum of a particle in the state of rest

whose invariant is defined by the expression (30).

Thus, the generalized momentum of the particle in an external field is not only invariant relative to the transformations at the transition from one reference system to another but also has an invariant representation in terms of the velocity of motion of the particle (30); at each point of space, the value of the invariant * I* is determined by the expression (35). This property has not only the representation of the proper momentum of the particle (the mechanical part), but also the generalized momentum of the particle in general.

Let us generalize this result to the case of representation of the generalized momentum of any systems and interactions, arguing that, regardless of the state (the motion) of the system, the generalized four-dimensional momentum always has an invariant representation

where ε и p are the energy and momentum of the system, respectively, and the invariant is determined by the modulus of sum of the components of the generalized momentum of the system

Let us represent the expression for the invariant

and divide it by

that is, we obtain the formula for the correspondence between the energy of the system

For example, the potential energy

Note, whatever is the dependence of the potential

Many well-known expressions of the potential energy of interaction with attractive fields have a repulsive component in the form of half the square of these attractive potentials—Kratzer [30], Lennard-Jones [31], Morse [32], Rosen [33] and others. Expression (41) justifies this approach, which until now is phenomenological or the result of an appropriate selection for agreement with experimental data.

The Hamiltonian

## 3. Equations of relativistic mechanics

### 3.1 Canonical Lagrangian and Hamilton-Jacoby equation

Let us use the parametric representation of the Hamilton action in the form [28].

where ds is the four-dimensional interval and

The functional that takes into account the condition of the invariant representation of the generalized momentum

where

where the four-dimensional momentum is represented in the form

Thus, the action is represented in the form

and the canonical Lagrangian of the system is given by

The correctness of the presented parametrization is confirmed by the obtained expressions for the generalized momentum and energy from the Lagrangian of the system in the form

which coincide with the initial representations of the generalized momentum and energy. Accordingly, the Lagrange equation of motion takes the form

If we multiply Eq. (50) by

If the invariant is clearly independent of time, then the energy ε is conserved and the equation of motion is represented in the form of the Newtonian equation

For a particle in an external field we have

Using the explicit form of the generalized momentum (32) with the accuracy of the expansion to the power of

where the velocity-dependent components of the force are present. In particular, the velocity-dependent force is present in the Faraday law of electromagnetic induction [34], which is absent in the traditional expression for the Lorentz force.

The Hamilton-Jacobi equation is represented in the form

and it reflects the invariance of the representation of the generalized momentum. The well-known representations of the Hamilton-Jacobi Eq. (8) also contain the differential forms of potentials—the components of the electric and the magnetic fields.

### 3.2 Motion of a charged particle in a constant electric field

Let us consider the motion of a charged particle with the mass m and charge

Let us represent the action

where

We find the solution from the condition

or

The well-known solution in the framework of the traditional theory [8] is the following:

In the ultrarelativistic limit

The electron velocity

### 3.3 Problem of the hydrogen-like atom

Let us consider the motion of an electron with the mass m and charge

Choosing the polar coordinates

Let us represent the action

where

We find trajectories from the condition

which results in the solution

The coefficient of the repulsive effective potential is essentially positive, that is,

The secular precession is found from the condition

whence, we obtain

that has the opposite sign as compared with the solution in [8]. The reason for the discrepancy of the sign is the unaccounted interaction of the self-momentum with the rotating field, that is, the spin-orbit interaction.

## 4. Equations of the relativistic quantum mechanics

Using the principle of the invariant representation of the generalized momentum

it is possible to compose the corresponding equation of the relativistic quantum mechanics by representing the energy and momentum variables by the corresponding operators

and

The case of conservative systems, when any energy losses or sources in space are absent, corresponds to the relation

For the charged particle in an external field with an invariant in the form of (30), the equations will take the form

For stationary states we obtain

Rewriting the equations taking into account the formulas of the classical correspondence (40), we will obtain the equations for the wave function in the traditional representation

the first of which formally coincides with the Schrödinger equation for the wave function of stationary states.

For the action function S associated with the wave function by the representation

which represents the exact classical correspondence instead of the quasiclassical approximation [12]. Note, the equations similar to (78) also follow from the Eq. (46) in [12] if we demand for an exact correspondence and equate to zero the real and imaginary parts.

### 4.1 Particle in the one-dimensional potential well

Let us consider the particle of mass

From the first equation of system (70) we have

Then,

where

In the three-dimensional case, the bound state with the energy

The solution of this simple example is fundamental and accurately represents the uncertainty principle

### 4.2 Penetration of a particle through a potential barrier

Let us consider the problem of penetration of a particle through the rectangular potential barrier [23] with the height

and at

where

For the problem of the passage of a particle with energy

and if the particle energy does not exceed the potential barrier, then the transmission coefficient is zero, regardless of the height of the barrier and not have. In this case, there is no contradiction similar to the Klein paradox.

### 4.3 Charged particle in a magnetic field

The vector potential of a uniform magnetic field

where

In this form, the Eq. (87) does not have a finite solution depending on the variable

Or

From (89) we have for the energy levels

where

If an electron is excited by a magnetic field from a state of rest, then

or

From (92) for a magnetic flux quantum we have

We get the same results when solving the Hamilton-Jacobi equation.

### 4.4 Particle in the field with Morse potential energy

We determine the energy levels for a particle moving in a field with a potential

According to (41), for the potential energy of interaction

Schrödinger equation takes the form

Following the procedure for solving Eq. (95) in [12], introducing a variable (taking values in the interval [0,

We get

Given the asymptotic behavior of function

equation of degenerate hypergeometric function (Kummer function).

A solution satisfying the finiteness condition for

In accordance with (96) and (99), we obtain values for energy levels

For the binding energy in the ground state

Because parameter

The interaction constant

We emphasize that despite the fact that the potential energy for a stationary particle

### 4.5 Problem of the hydrogen-like atom

The motion of a charged particle in the Coulomb field can be described as a motion in the field of an atomic nucleus (without the spin and magnetic moment) with the potential energy

In spherical coordinates, Eq. (70) for the wave function takes the form

Separating the variables

and introducing the notations [12]

(only the positive root is taken for

where

The solution of Eq. (88) formally coincides with the well-known Fuse solution for the molecular Kratzer potential in the form

where the radial quantum number

For the ground state with the

without any restrictions for the value of

In this case, the obtained fine splitting is in no way connected with the spin-orbit interaction and is due to the relativistic dependence of the mass on the orbital and radial velocity of motion, which results to the splitting of the levels.

### 4.6 Dirac equations

In the standard representation, the Dirac equations in compact notation for a particle have the form [21].

In addition, for the particle in an external field they can be represented in theorm

By writing the wave equations for the wave functions, we obtain

where we used the properties of the Pauli matrices. It is easy to verify that the functions

In the case of a stationary state, the standard representation of the wave Eq. (110) has the form

### 4.7 Dirac equations solution for a hydrogen-like atom

For a charge in a potential field with the central symmetry [23], we have

After substituting (96) into (95), we obtain

Let us represent the functions f and

where

Substituting (116) into the Eq. (117), for the sum and difference of the equations we have

Close to

Then

Forming equations of the second order and solving with respect to

With allowance for (121), the solution of these equations is

where

From expressions (117), we obtain

where

and taking into account the obtained values of χ, we finally have

where the principal quantum number

without any limitations for the value of

In the resulting formula (126), the order of sequence of the fine splitting levels is inverse relative to the order of sequence in the well-known Sommerfeld-Dirac formula. If to compare the expansions in a series in the degree of the fine-structure constant of two formulas

then the difference will be equal to

where the last term is the expression for the spin-orbit interaction energy. Thus, to obtain the true value of the energy levels of the hydrogen atom, it is necessary to add the energy of the spin-orbit interaction in formula (126) in the form (130). This is completely justified, because such an interaction was not initially included in Eq. (115) and was not reflected in the final result.

## 5. Conclusion

The principle of invariance is generalized and the corresponding representation of the generalized momentum of the system is proposed; the equations of relativistic and quantum mechanics are proposed, which are devoid of the above-mentioned shortcomings and contradictions. The equations have solutions for any values of the interaction constant of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus Z > 137. The equations are applicable for different types of particles and interactions.

Based on the parametric representation of the action and the canonical equations, the corresponding relativistic mechanics based on the canonical Lagrangian is constructed and the equations of motion and expression are derived for the force acting on the charge moving in an external electromagnetic field.

The matrix representation of equations of the characteristics for the action function and the wave function results in the Dirac equation with the correct enabling of the interaction. In this form, the solutions of the Dirac equations are not restricted by the value of the interaction constant and have a spinor representation by scalar solutions of the equations for the action function and the wave function.

The analysis of the solutions shows the full compliance with the principles of the relativistic and quantum mechanics, and the solutions are devoid of any restrictions on the nature and magnitude of the interactions.

The theory of spin fields and equations for spin systems will be described in subsequent works.

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