_{1}

^{*}

Using a novel wave equation, which is Galileo invariant but can give precise results up to energies
as high as
mc
^{2}
, 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 Schr
ö
dinger equation.

Quantum mechanics triumphed when physicists learned to describe the quantum states of the electrons in the atoms by solving the Schrödinger equation [^{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 Grave de Peralta equation is explored [

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 [

K = p 2 2 m , p = m V . (1)

Then, substituting K and p by the following energy and momentum quantum operators [

E ^ = K ^ = i ℏ ∂ ∂ t , p ^ = − i ℏ ∂ ∂ x . (2)

In Equation (2), ℏ is the Plank constant (h) divided by 2π, results [

i ℏ ∂ ∂ t ψ S c h ( x , t ) = − ℏ 2 2 m ∂ 2 ∂ x 2 ψ S c h ( x , t ) . (3)

However, Equation (1) does not give the correct relation between K and p when the particle moves at faster speeds. Correspondingly, the Schrödinger equation (Equation (3)) is not Lorentz invariant but Galileo invariant [

E 2 − m 2 c 4 = p 2 c 2 ⇔ ( E + m c 2 ) ( E − m c 2 ) = p 2 c 2 . (4)

And:

E = γ V m c 2 , p = γ V m V . (5)

Here, E = K + mc^{2} is the total relativistic energy of the free particle, and [

γ V = 1 1 − V 2 c 2 . (6)

One can then formally proceed as it is done for obtaining the 1D Schrödinger equation, and use Equation (2) for assigning the temporal partial derivative operator to E in the first expression of Equation (4) [

1 c 2 ∂ 2 ∂ t 2 ψ K G ( x , t ) = ∂ 2 ∂ x 2 ψ K G ( x , t ) − m 2 c 2 ℏ 2 ψ K G ( x , t ) . (7)

The Klein-Gordon equation is Lorentz invariant and describes a free quantum particle with mass m and spin-0 [^{2}) is always different than 0 for E > 0; consequently, Equation (4) and the following algebraic equation are equivalents for E > 0:

( E − m c 2 ) = p 2 ( γ V + 1 ) m (8)

Each member of Equation (8) is just a different expression of the relativistic kinetic energy of the particle [

i ℏ ∂ ∂ t ψ K G + ( x , t ) = − ℏ 2 ( γ V + 1 ) m ∂ 2 ∂ x 2 ψ K G + ( x , t ) + m c 2 ψ K G + ( x , t ) . (9)

A simple substitution in Equation (7) and Equation (9) shows that the following plane wave is a solution of both equations for E > 0:

ψ K G + ( x , t ) = e i ℏ ( p x − E t ) . (10)

The plane wave ψ_{KG}_{+} has an unphysical phase velocity equal to c^{2}/V > c [

ψ ( x , t ) = ψ K G + e i w m t , w m = m c 2 ℏ . (11)

Such that ψ has a phase velocity smaller than c [

ψ ( x , t ) = e i ℏ ( p x − K t ) . (12)

Substituting ψ given by Equation (11) in Equation (9) results in the 1D Grave de Peralta equation [

i ℏ ∂ ∂ t ψ ( x , t ) = − ℏ 2 ( γ V + 1 ) m ∂ 2 ∂ x 2 ψ ( x , t ) . (13)

Equation (13) clearly coincides with the Schrödinger equation at low particle’s speeds. Moreover, a positive probability density can be defined for the solutions of Equation (13) by analogy of how it is defined for the solutions of the Schrödinger equation and, like the Schrödinger equation, Equation (13) is Galilean invariant for observers traveling at low speeds respect to each other [

i ℏ ∂ ∂ t ψ ( x , t ) = − ℏ 2 [ γ V ( x ) + 1 ] m ∂ 2 ∂ x 2 ψ ( x , t ) + U ( x ) ψ ( x , t ) . (14)

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 = E – mc^{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 [_{V}, and V^{2} are constants [

ψ ( x , t ) = X K ( x ) e − i ℏ Ę t , Ę = K + U (15)

X_{K} is a solution of the following equation [

d 2 d x 2 X K ( x ) + κ 2 X K ( x ) = 0 , κ = p ℏ . (16)

And [

κ = p ℏ = 1 ℏ ( γ V + 1 ) m K = 1 ℏ ( γ V + 1 ) m ( Ę − U ) . (17)

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:

K = ℏ 2 κ 2 ( γ V + 1 ) m . (18)

As expected, when γ_{V} ~ 1, Equation (18) gives the non-relativistic values of the particle’s energies at low speeds, K ~ ℏ 2 κ 2 / ( 2 m ) [

γ V 2 = 1 + ( λ C λ ) 2 , λ C = h m c , λ = 2 π κ . (19)

In Equation (19), λ_{C} is the Compton wavelength associate to the mass of the particle [

K = ℏ 2 κ 2 [ 1 + 1 + ( λ C λ ) 2 ] m . (20)

As expected, Equation (20) match the non-relativistic expression of the particle’s kinetic energy when p = h/λ is very small because λ ≫ λ C . However, in each region where the value of U is constant, the values of K and then Ę = K + U calculated using Equation (20) are smaller than the ones calculated using the Schrödinger equation.

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 [

i ℏ ∂ ∂ t ψ ( r , t ) = − ℏ 2 [ γ V ( r ) + 1 ] m ∇ 2 ψ ( r , t ) + U ( r ) ψ ( r , t ) . (21)

In Equation (21), γ_{V} and V^{2} depend only on the radial variable (r) because the potential is a central potential. In spherical coordinates, the Laplacian operator in Equation (21) is given by the following expression [

∇ 2 ψ = 1 r ∂ 2 ∂ r 2 ( r ψ ) + 1 r 2 ∇ θ , φ 2 ψ . (22)

In Equation (22):

∇ θ , φ 2 = 1 sin θ ∂ ∂ θ ( sin θ ∂ ∂ θ ) + 1 sin 2 θ ∂ 2 ∂ φ 2 . (23)

Using Equation (22) and Equation (23) allows for rewriting Equation (21) in the following way [

i ℏ ∂ ∂ t ψ = − ℏ 2 [ γ V ( r ) + 1 ] m r ∂ 2 ∂ r 2 ( r ψ ) − ℏ 2 [ γ V ( r ) + 1 ] m r 2 ∇ θ , φ 2 ψ + U ( r ) ψ . (24)

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 [

i ℏ ∂ ∂ t ψ ( θ , φ ) = − ℏ 2 ( γ V + 1 ) m r 2 ∇ θ , φ 2 ψ ( θ , φ ) . (25)

The explicit absence of a potential in Equation (25) determines that it has solutions with constant values of Ę, K, γ_{V} and V^{2} [_{V} and V^{2} depending on r. Looking for a solution of Equation (24) as in Ref. [

ψ ( r , θ , φ , t ) = R ( r ) Ω ( θ , φ ) e i ℏ Ę t . (26)

Results:

∇ θ , φ 2 Ω ( θ , φ ) = η Ω ( θ , φ ) . (27)

And:

1 r d 2 d r 2 ( r R ) + [ γ V ( r ) + 1 ] m r 2 ℏ 2 [ Ę − U ( r ) ] R = − η R r 2 . (28)

Equation (27) is the well-known equation for the spherical harmonic functions [

Ω l , m ( θ , φ ) = Y l ( m ) ( θ , φ ) ; η = l ( l + 1 ) ; l = 0 , 1 , 2 , ⋯ ; m = − l , − l + 1 , ⋯ , 0 , 1 , ⋯ , l . (29)

Here, Y l ( m ) are the spherical harmonic functions [

d 2 d r 2 χ ( r ) + [ γ V ( r ) + 1 ] m ℏ 2 [ Ę − W ( r ) ] χ ( r ) = 0. (30)

In Equation (30):

W ( r ) = [ U ( r ) + ℏ 2 [ γ V ( r ) + 1 ] m l ( l + 1 ) r 2 ] . (31)

The radial equation, Equation (30), is then formally identical to Equation (16) with:

κ ( r ) = p ( r ) ℏ = 1 ℏ [ γ V ( r ) + 1 ] m K ( r ) = 1 ℏ [ γ V ( r ) + 1 ] m [ Ę − W ( r ) ] . (32)

At low particle speeds, γ_{V}(r) ~ 1, thus Equation (30) coincide with the radial equation that can be obtained when solving the same problem using the Schrödinger equation [_{V} depends on r; therefore, in general, the solutions of Equation (30) are different than the solutions of the radial equation for the Schrödinger equation. A notable exception is the infinite spherical well problem for l = 0 where U(r) is given by the following expression [

U ( r ) = { 0 , r < r o U o → + ∞ , r ≥ r o (33)

In this case W ( r ) ≡ 0 ; thus K, γ_{V}, and V^{2} are constant inside the well. Equation (30) can then be solved as it is done for the Schrödinger equation. Consequently [

Ę n = K n = n 2 h 2 ( γ V + 1 ) m D o 2 . (34)

In Equation (34), D_{o} = 2r_{o} and n is a positive integer number. From Equation (34) and the relativistic equation K = ( γ V − 1 ) m c 2 follows that:

γ V 2 = 1 + n 2 ( λ C D o ) 2 , λ C = h m c . (35)

Substituting γ_{V} given by Equation (35) in Equation (34) results:

Ę n = n 2 [ 1 + 1 + ( n λ C D o ) 2 ] ( λ C D o ) 2 m c 2 . (36)

As expected, γ_{V} ~ 1 when n = 1 and D o ≫ λ C ; thus, Equation (36) coincides with the energies of the infinite spherical well calculated using the Schrödinger equation [_{C}, the minimum particle energy is quasi-relativistic; therefore, Equation (36) must be used. For instance, γ V 2 = 2 , V ~ 0.7c, and K ~ 0.4mc^{2} when Equation (35) is evaluated for n = 1 and D_{o} = λ_{C}. However, γ V 2 = 5 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 1 / 8 λ C 3 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 may also be true for a black hole and the whole universe at the beginning of the Big Bang. This is consistent, for instance, with the confinement of an electron in the Hydrogen atom because for an electron λ_{C} ~ 2.4 × 10^{−3} nm, which is ~ 20 times smaller than the Bohr radius of the Hydrogen atom, r_{B} ~ 5.3 × 10^{−2} nm [

In the Hydrogen atom or in highly ionized atoms with a single electron, U(r) is the Coulomb potential [

U ( r ) = U C ( r ) = − e 2 4 π ε o Z r . (37)

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) with the electron mass, m_{e}, substituted by the reduced mass of the electron, μ = ( m e m n ) / ( m e + m n ) , i.e.:

d 2 d r 2 χ ( r ) + [ γ V ( r ) + 1 ] μ ℏ 2 [ Ę − W C ( r ) ] χ ( r ) = 0. (38)

In Equation (38):

W C ( r ) = [ U C ( r ) + ℏ 2 [ γ V ( r ) + 1 ] μ l ( l + 1 ) r 2 ] . (39)

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 [_{V} from Equation (38) and Equation (39) by making:

[ γ V ( r ) + 1 ] μ ℏ 2 = K + 2 μ c 2 c 2 ℏ 2 = [ Ę − U C ( r ) ] + 2 μ c 2 c 2 ℏ 2 . (40)

Using Equation (40) then allows for rewriting Equation (38) in the following way:

{ − ℏ 2 2 μ d 2 d r 2 χ ( r ) − [ Ę − U C ( r ) ] χ ( r ) + ℏ 2 2 µ l ( l + 1 ) r 2 χ ( r ) } − 1 2 μ c 2 [ Ę − U C ( r ) ] 2 χ ( r ) = 0. (41)

The term between braces in Equation (41) coincides with the radial equation that should be solved when using the Schrödinger equation [

ζ = 1 ℏ − 2 μ Ę . (42)

For bound states, Ę < 0; therefore, ζ is real. Using Equation (42) allows for rewriting Equation (41) in the following way:

1 ζ 2 d 2 d r 2 χ ( r ) = { [ 1 − μ e 2 2 π ε o ℏ 2 ζ Z ( ζ r ) + l ( l + 1 ) ( ζ r ) 2 ] − [ α 2 Z 2 ( ζ r ) 2 − ℏ ζ μ c α Z ( ζ r ) + ( ℏ 2 μ c ) 2 ζ 2 ] } χ ( r ) . (43)

where α is the fine-structure constant [

α = 1 4 π ε o e 2 ℏ c ~ 1 / 137 . (44)

It is convenient to rewrite Equation (43) as:

1 ζ 2 d 2 d r 2 χ ( r ) = { [ 1 − ( ℏ ζ 2 μ c ) 2 ] − [ ( μ e 2 2 π ε o ℏ 2 ζ − α ℏ ζ μ c ) Z ( ζ r ) ] + l ( l + 1 ) − α 2 Z 2 ( ζ r ) 2 } χ ( r ) . (45)

So that introducing the new variables:

ρ ≡ ζ r , ρ o ≡ ( μ e 2 2 π ε o ℏ 2 ζ − α ℏ ζ μ c ) Z , ρ 1 ≡ [ 1 − ( ℏ ζ 2 μ c ) 2 ] . (46)

Allows for rewriting Equation (45) in the following way:

d 2 d ρ 2 χ ( ρ ) = [ ρ 1 − ρ o ρ + l ( l + 1 ) − α 2 Z 2 ρ 2 ] χ ( ρ ) . (47)

If the electron was free and moving slowly with kinetic energy K = Ę, then its linear momentum would be ℏ ζ ≪ μ c . In this limit, one can approximate Equation (46) in the following way [

ρ ≡ ζ r , ρ o ~ μ e 2 Z 2 π ε o ℏ 2 ζ , ρ 1 ~ 1. (48)

Using Equation (48), and considering that for the Hydrogen atom α 2 Z 2 ≪ 1 , allows for approximating Equation (47) in the following way [

d 2 d ρ 2 χ ( ρ ) = [ 1 − ρ o ρ + l ( l + 1 ) ρ 2 ] χ ( ρ ) . (49)

which is the equation that is solved for the Hydrogen atom when using the Schrödinger equation [

ρ o = 2 n , n = 1 , 2 , 3 , ⋯ (50)

From Equation (50), Equation (48), and Equation (42) then follows for Z = 1 the following well-known result [

Ę n , S c h = − [ μ 2 ℏ 2 ( e 2 4 π ε o ) 2 ] 1 n 2 . (51)

However, each of the three terms in the right side of Equation (47) contains a different quasi-relativistic correction to the radial equation of hydrogen-like atoms. Nevertheless, one can try to solve the quasi-relativistic Equation (47) as Equation (49) is solved [

d 2 d ρ 2 χ ( ρ ) = ρ 1 χ ( ρ ) . (52)

which general solution is:

χ ( ρ ) = A e − ρ 1 ρ + B e ρ 1 ρ . (53)

But ρ 1 > 0 ; therefore, B must be null, so for large ρ:

χ ( ρ ) ~ A e − ρ 1 ρ . (54)

On the other hand, when ρ → 0 the centrifugal term dominates; approximately, then:

d 2 d ρ 2 χ ( ρ ) = l ( l + 1 ) − α 2 Z 2 ρ 2 χ ( ρ ) . (55)

Which general solution is:

χ ( ρ ) = C ρ 1 2 [ 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ] + D ρ 1 2 [ 1 − ( 1 + 2 l ) 2 − 4 α 2 Z 2 ] . (56)

Therefore, D must be null, so for small ρ:

χ ( ρ ) ~ C ρ 1 2 [ 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ] . (57)

As expected, if the quasi-relativistic corrections are very small, then Equation (54) and Equation (57) reduces to the ones obtained when using the Schrödinger equation [

χ ( ρ ) ≡ τ ( ρ ) ρ 1 2 [ 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ] e − ρ 1 ρ . (58)

From Equation (58) and Equation (47) then follow that τ ( ρ ) is a solution of the following equation:

ρ d 2 d ρ 2 τ ( ρ ) + [ 1 − ( 2 ρ 1 ) ρ + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ] d d ρ τ ( ρ ) + [ ρ o − ρ 1 ( 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ) ] τ ( ρ ) = 0. (59)

Again, as expected, if the quasi-relativistic corrections are very small, then Equation (59) reduces to the one obtained when using the Schrödinger equation [

τ ( ρ ) = ∑ j = 0 j max a j ρ j . (60)

And substituting Equation (60) in Equation (59) results:

a j + 1 = ρ 1 [ 2 j + ( 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ) ] − ρ o ( j + 1 ) [ j + ( 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ) ] a j . (61)

Evaluating Equation (61) for j = j max and making a j max + 1 = 0 , results:

ρ o ρ 1 = [ 2 j + ( 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ) ] . (62)

As expected, if the quasi-relativistic corrections are very small, then Equation (62) reduces to Equation (50) with n = j + l + 1 [

ρ o = [ 2 n + Δ ( l , Z ) ] ρ 1 . (63)

In Equation (63):

Δ ( l , Z ) = [ ( 1 + ( 1 + 2 l ) 2 − 4 α 2 Z 2 ) − 2 ( l + 1 ) ] . (64)

Substituting ρ o and ρ 1 given by Equation (46) in Equation (63), solving the resulting equation for ζ , and using Equation (42) allows for obtaining an exact analytical expression for Ę, which now depends not only on the principal quantum number n, but also on the angular quantum number l, and Z . For instance, assuming that the quasi-relativistic corrections included in ρ o and ρ 1 do not need to be accounting for because they are too small, the effect of the quasi-relativistic correction included in the centrifugal term in Equation (63) is quantified by the following equation:

Ę n , l = − [ μ 2 ℏ 2 ( e 2 2 π ε o ) 2 ] Z 2 [ 2 n + Δ ( l , Z ) ] 2 . (65)

As expected, if α was null and Z = 1, then Equation (65) would be identical to Equation (51). However, Δ ( l , Z ) < 0 and | Δ ( l , Z ) | increases when Z increases. Therefore, for n > 1 and l > 0, the degeneration of Ę_{n} given by Equation (51) is broken by the quasi-relativistic correction Δ ( l , Z ) . This effect is more pronounced for heavy elements. In addition, as shown in

schematic of the calculated values of Ę n , l − Ę n in eV, where Ę_{n}_{,l} and Ę_{n} were evaluated using Equation (65) with Z = 1 and Equation (51), respectively. In all cases, stabilizing negative quasi-relativistic corrections to the non-relativistic energies were obtained. This is because the negative contribution of −α^{2}Z^{2} in the numerator of the centrifugal term in Equation (47).

In columns 2 and 3 in _{n}_{,l} (in eV) that were calculated using Equation (65) and Equation (63), respectively. The difference between the approximated values (Equation (65)) and the exact values (Equation (63)) of Ę_{n}_{,l} are ~0.01 meV; thus smaller than the exact values of Ę n , l = 0 − Ę n reported in the third column of _{n} [

Ę n , l ~ Ę n , S c h Z 2 { 1 − α 2 n 2 [ 3 4 − n l + 1 2 ] } . (66)

where Ę_{n}_{,l,Sch} given by Equation (51) correspond to the Hydrogen energies calculated using the Schrödinger equation. In correspondence with this, one can show that the values of Ę_{n}_{,l} calculated using Equation (65) are approximately equal to:

Ę n , l ~ Ę n , S c h Z 2 ( 1 + 2 α 2 n ( 2 l + 1 ) ) . (67)

Indeed, Equation (65) can be rewritten as:

Ę n , l = − μ c 2 α 2 Z 2 1 [ 2 n + Δ ( l , Z ) ] 2 . (68)

(n,l) | Equation (65) | Equation (63) | E_{nl} − E_{n} (meV) | (n',l') → (n,l) | E_{n’l’} − E_{n}_{,l} | λ (nm) |
---|---|---|---|---|---|---|

(1,0) | −13.5997 | −13.5992 | −0.90526 | (2,1) → (1,0) | 10.1996 | 121.558 |

(2,0) | −3.39975 | −3.39972 | −0.147102 | (3,1) → (2,0) | 1.88879 | 656.422 |

(2,1) | −3.39963 | −3.3996 | −0.0264008 | (3,0) → (2,1) | 1.88863 | 656.477 |

(3,0) | −1.51097 | −1.51097 | −0.046938 | (3,2) → (2,1) | 1.88867 | 656.462 |

(3,1) | −1.51094 | −1.51093 | −0.0111749 | (3) → (2,1) | 1.88863 | 656.4695 |

(3,2) | −1.51093 | 1.51092 | −0.00402301 |

Then Equation (67) can be obtained from Equation (68) using the following approximated relations:

1 [ 2 n + Δ ( l , Z ) ] 2 ~ 1 ( 2 n ) 2 [ 1 − Δ ( l , Z ) n ] , Δ ( l , Z ) ~ − 2 α 2 2 l + 1 . (69)

This is an important result: the quasi-relativist energies calculated using the Grave de Peralta equation corresponds to the sum of the non-relativistic energies calculated using the Schrödinger equation plus the relativistic corrections to the kinetic energy. Consequently, these energies do not include the Darwin energy term [_{n}_{,l} reported in the third column of _{1} = 656.422 nm spectral line produced by the (3,1) to (2,0) atomic transition and the λ_{2} = 656.4695 nm spectral line, which was estimated as in the middle of the spectral

(In nm) | Experimental | Calculated |
---|---|---|

α-Lyman (λ in nm) | 121.567 | 121.558 |

α-Lyman (Δλ in nm) | 0.006 | No |

α-Balmer (λ in nm) | 656.279 | 656.422 |

α-Lyman (Δλ in meV) | 0.04 | 0.16 |

lines corresponding to the atomic transitions (3,0) to (2,1) and (3,2) to (2,1). This corresponds to a Balmer’s α-doublet separation of ∆λ ~ 0.048 nm or ∆E ~ 0.16 meV. Nevertheless, as shown in

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.

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

de Peralta, L.G. (2020) Quasi-Relativistic Description of Hydrogen-Like Atoms. Journal of Modern Physics, 11, 788-802. https://doi.org/10.4236/jmp.2020.116051