Quasi-Relativistic Description of Hydrogen-Like Atoms

Using a novel wave equation, which is Galileo invariant but can give precise results up to energies as high as mc, exact quasi-relativistic quantum mechanical solutions are found for the Hydrogen atom. It is shown that the exact solutions of the Grave de Peralta equation include the relativistic correction to the non-relativistic kinetic energies calculated using the Schrödinger equation.


Introduction
Quantum mechanics triumphed when physicists learned to describe the quantum states of the electrons in the atoms by solving the Schrödinger equation [1] [2] [3] [4] [5]. However, the Schrödinger equation is not Lorentz invariant but Galilean invariant [6] [7]; therefore, a relativistic quantum mechanics cannot be based on the Schrödinger equation. A fully relativistic quantum theory requires to be funded on equations that are valid for any two observers moving respect to each other at constant velocity [8] [9]. In contrast, the Galilean invariance of the Schrödinger equation means that two such observers will only agree in the adequacy of the Schrödinger equation for describing the movement of a massive free quantum particle when the relative speed between the observers is much smaller than the speed of the light in the vacuum (c). In practice, this is not a terrible limitation of the Schrödinger equation because up to today humans have been only able to travel at speeds much smaller than c. This is one of the principal reasons why the Schrödinger equation is still relevant almost 100 years after its discovery. Moreover, there is another important limitation of the Schrödinger equation: it describes a particle in which linear momentum (p) and kinetic energy (K) are related by a classical relation that is not valid at relativistic speeds [6] [7] [8] [9] [10]. Nevertheless, wave mechanics triumphed when Schrödinger, using his equation, was able to reproduce the results previously obtained by Bohr for the energies of the bound states of the electron in the Hydrogen atom. This was possible because the electron in the Hydrogen atom moves at non-relativistic energies [1] [2] [3] [4] [5]. Rigorously, the number of particles may be not constant in a fully relativistic quantum theory [7] [8] [9]. This is because when the sum of the kinetic and the potential (U) energy of a particle with mass m equals the energy associate to the mass of the particle, i.e. 2 Ę K U mc = + = , then a second particle with the same mass could be created from Ę. Consequently, the number of particles is constant when . This is what happens in atoms and molecules; thus, this explains why the results obtained using the Schrödinger equation are a good first approximation in chemistry applications [5]. In between the Galilean invariant Schrödinger equation and the fully relativistic quantum mechanics, there is a quasi-relativistic region where Ę < mc 2 but Ę is so large that it is necessary to use an equation that describes a particle having a relativistic relation between p and K. In this work, the use of the

The Grave de Peralta Equation
Formally, the one-dimensional (1D) Schrödinger equation for a free quantum particle with mass m can be obtained from the classical relation between K and p for a free particle when its speed (V) is much smaller than the c [ The Klein-Gordon equation is Lorentz invariant and describes a free quantum particle with mass m and spin-0 [8] [9]. In contrast to the Schrödinger equation, a second-order temporal derivative is present in Equation (7). This determines that Equation (7) has solutions with positive and negative energy values while Equation (3) has only solutions with positive energies, which is in correspondence with K having only positive values in Equation (1) but E having positive and negative values in Equation (4). The factor (E + mc 2 ) is always different than 0 for E > 0; consequently, Equation (4) and the following algebraic equation are equivalents for E > 0: Each member of Equation (8) is just a different expression of the relativistic kinetic energy of the particle [7]. Assigning the temporal partial derivative operator in Equation (2) to E in Equation (8) results in the following differential equation [11]: A simple substitution in Equation (7) and Equation (9) shows that the follow- The plane wave ψ KG+ has an unphysical phase velocity equal to c 2 /V > c [7] [11]. However, one can look for a solution of Equation (9) of the following form: Such that ψ has a phase velocity smaller than c [7] [11]; thus: Substituting ψ given by Equation (11) in Equation (9) Equation (13) (13) is Galilean invariant for observers traveling at low speeds respect to each other [7]. Despite this, Equation (13) can be used for obtaining precise solutions of very interesting quantum problems at quasi-relativistic energies [7] [11], where a particle moves at so large speeds that it is necessary to use the correct relativistic relation between p and K, but where the particle should not be moving too fast so that the number of particles remains constant. When the particle moves through a 1D piecewise constant potential U(x), Equation (13) should be generalized in the following way [11]: Often, one looks for solutions of Equation (14) corresponding to a constant value of the energy Ę = K + U, where Ę is not the total relativistic energy of the particle (E) but Ę = Emc 2 . At quasi-relativistic energies, the number of particles is constant; therefore, Ę is constant whenever E is constant. For a 1D piecewise constant potential Ę, K, γ V , and V 2 are constants in each x-region where U is constant. In contrast to Ę, however, K, γ V and V 2 have a discontinuity wherever U(x) has one. Consequently, in Equation (14) γ V is a function of x because, in general, the square of the particle's speed (V 2 ) depends on the position [11]. Nevertheless, for 1D piecewise constant potentials, one can look for a solution of Equation (14) And [11]: Consequently, κ and X K are not determined by the values of Ę but by the values of K = Ę -U. Once the allowed values of κ are determined from Equation (16) and the boundary conditions, the allowed values of the relativistic kinetic energy of the particle K = Ę -U are given by: As expected, when γ V ~ 1, Equation (18) In Equation (19), λ C is the Compton wavelength associate to the mass of the particle [8] [10], and λ is the De Broglie wavelength of the wavefunction given by Equation (7) and Equation (10) (18) allows obtaining an analytical expression of the precise quasi-relativistic kinetic energy of the particle: As expected, Equation (20)

Movement in a Central Potential
A quantum state of a particle with mass m moving at quasi-relativistic energies in a central potential, U(r), is a solution of the following 3D Grave de Peralta equation [7]: In Equation (21) Using Equation (22) and Equation (23) allows for rewriting Equation (21) in the following way [7]: The second term of the right size of Equation (24) corresponds to the rotational energy of the particle. For a quantum rotor, which describes a particle moving in a sphere, r is constant [2] [5]. This allows for simplifying Equation (24) in the following way [7]: The explicit absence of a potential in Equation (25)    Here, ( )  [5]. However, at quasi-relativistic energies, γ V depends on r; therefore, in general, the solutions of Equation (30) As expected, γ V ~ 1 when n = 1 and o C D λ  ; thus, Equation (36) coincides with the energies of the infinite spherical well calculated using the Schrödinger equation [4]. In contrast, when the diameter of the well is close to λ C , the minimum particle energy is quasi-relativistic; therefore, Equation (36) must be used. For instance, 2 2 V γ = , V ~ 0.7c, and K ~ 0.4mc 2 when Equation (35) is evaluated for n = 1 and D o = λ C . However, 2 5 V γ = and K ~ 1.2mc 2 when n = 1 and D o = λ C /2. The number of particles may not be constant at these energies. Consequently, the Grave de Peralta equation establishes a fundamental connection between quantum mechanics and especial theory of relativity: no single particle with mass can be confined in a volume much smaller than 3 1 8 C λ because when this occurs, K > mc 2 and the number of particles may not be constant anymore; therefore, a single point-particle with mass cannot exist. Point-particles with mass can only exist in fully relativistic quantum field theories where the number of particles is not constant. This is true for an electron, a quark, and probably  [5]. It should be noted that strictly speaking, the problem corresponding to the potential defined by Equation (33) is a relativistic problem because and thus the number of particles may not be constant. Nevertheless, the non-relativistic and quasi-relativistic infinite well problems could be considered approximations to the problem of a quantum particle absolutely trapped in a finite region. This is because for obtaining Equation (34) and Equation (36) the infinitude of the potential is only used for arguing that χ(r) should be null everywhere except inside of the well, thus assigning null boundary conditions to Equation (30).

Hydrogen-Like Atoms
In the Hydrogen atom or in highly ionized atoms with a single electron, U(r) is the Here, e is the electron charge, Z is the atomic number, and ε o is the electric permittivity of vacuum. Therefore, the radial equation corresponding to the quasi-relativistic states of the electron in a hydrogen-like atom with a nucleus of mass m n is given by the Equation (30) In Equation (38): As expected, when the electron moves slowly ( V c  ) then γ V ~ 1; therefore, Equation (38) reduces to the radial equation of a hydrogen-like atom obtained using the Schrödinger equation [4]. Using Equation (5), it is possible to eliminate γ V from Equation (38) and Equation (39) by making: Using Equation (40) then allows for rewriting Equation (38) in the following way: For bound states, Ę < 0; therefore, ζ is real. Using Equation (42) allows for rewriting Equation (41) in the following way: where α is the fine-structure constant [ However, each of the three terms in the right side of Equation (47)  .
This corresponds to a Balmer's α-doublet separation of ∆λ ~ 0.048 nm or ∆E ~ 0.16 meV. Nevertheless, as shown in Table 2, this value is four times larger than the experimental value [14], which demonstrates the need for including in the calculation both the Darwin and the spin-orbit contributions.

Conclusion
It has been shown how to solve the Grave de Peralta equation for a charged quantum particle with mass and spin-0, which is moving in a Coulomb potential or contained in a spherical infinite well. The solutions were found following the same procedures and with no more difficulty than the corresponding to solving the same problems using the Schrödinger equation. Nevertheless, the solutions found in this work are also valid when the particle is moving with quasi-relativistic energies. For instance, it was shown that the energies of the electron in a Hydrogen atom, which were calculated by solving the Grave de Peralta equation, includes the relativistic Thomas correction. Moreover, the relativistic correction to the kinetic energy is just an approximation found using a perturbative approach while Equation (63) was exactly solved. In addition, it should be noted that Equation (41) is different than the radial equation obtained using the Schrödinger equation. The author is currently working on solving Equation (41).
This will allow to obtain more precise expressions for the atomic orbitals currently used in numerous ab initio computer packages dedicated to computer calculations in physical-chemistry and atomic and solid-state physics.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.