Exact and Approximate Solutions of Dirac–Morse Problem in Curved Space-Time

In this work, we analyze the Dirac–Morse problem with spin and pseudo-spin symmetries in deformed nuclei. So, we consider the Dirac equation with the scalar U ( r ) and vector V ( r ) Morse-type potentials and tensor Hellmann-type potential in curved space-time whose line element is of type ds 2 = ( 1 + α 2 U ( r )) 2 ( dt 2 − dr 2 ) − r 2 d θ 2 − r 2 sin 2 θ d φ 2 . From the effective tensor potential A ef f ( r ) = λ/ r + α 2 λ U ( r )/ r + A ( r ) , that contain terms of spin-orbit coupling, line element and electromagnetic ﬁeld, we analyze dirac’s spinor in two ways: (i) in the ﬁrst, we solve the problem approximately considering A ef f ( r ) not null; (ii) in the second analysis, we obtain exact solutions of radial spinor and eigenenergies considering A ef f ( r ) = 0. In both cases, we consider two types of coupling of vector and scalar potentials, with spin symmetry for V ( r ) = U ( r ) and pseudo-spin symmetry for V ( r ) = − U ( r ) . We analyzed the effect of coupling the electromagnetic ﬁeld with the curvature of space in eigenenergies and radial spinor.


Introduction
Symmetries in Physics are very important and have several studies and applications in areas such as classical and quantum mechanics as well as in electromagnetism.Particularly in quantum mechanics we have the spin and pseudo-spin symmetry [1,2].The pseudo-spin symmetry manifest in exotic nuclei and is observed by the quasi-degenerescence between doublet single-particle states (n, l, j = l + 1/2) and (n − 1, l + 2, j = l + 3/2), with n, l and j being the radial, orbital and the total angular momentum quantum numbers, respectively.Doublet states can also be written as ( ñ, l, l±1/2).In 1997, Ginocchio [3] showed that the pseudo-spin symmetry arise in nucleons moving in a relativistic mean field, with the pseudo-angular momentum l = l +1 being the component of lower spinor [4].Subsequently, the analysis of pseudo-spin symmetry was amplified for deformed nuclei [5,6], thus it was shown that such symmetry also manifests itself in a nucleus with deformation in its structure, which are nuclei that may contain, for example, a halo of neutrons.For spin symmetry the doublets states are given by (n, l, j = l + 1/2) and (n, l, j = l − 1/2).In the relativistic regime, spin and pseudo-spin symmetries are introduced in the Dirac equation through of coupling of the scalar U (r ) and vectorial V (r ) potentials as follows [7]: (i) (r ) = U (r ) − V (r ) = constant for spin symmetry and (ii) (r ) = U (r ) + V (r ) = constant for pseudo-spin symmetry.
Among the potentials already applied with spin and pseudo-spin symmetry are the Morse-type potentials that were investigated with Coulomb-type tensor potential using Pekeris approximation [8] in the works [9][10][11][12].Also, the Dirac equation with Yukawa potential and with Coulomb tensor potential was studied, using Pekeris approximation, by Ikhdair, Maghsoodi, and collaborators [13,14].Another interesting potential that includes the Coulomb plus Yukawa potentials is known as Hellmann potential.It was used to model interaction electron-core [15,16], as well as atomic scattering [17,18], and recently it was applied to the Dirac equation with spin and pseudo-spin symmetry [19,20].All these works have something in common, which was to have Address(es) of author(s) should be given analyzed the problem in an approximate way, this was necessary due to the coupling of the cited potentials with the spin-orbit coupling term, also known as pseudo-centrifugal potential, which is proportional to 1/r 2 .The presence of this term makes it impossible to exatcly analyze spinor and eigenenergies.
The applications of spin and pseudo-spin symmetries mentioned in the above paragraph consider spherical nuclei.However, pseudo-spin symmetry is not only manifested in spherical nuclei, which are nuclei that have a perfectly spherical shape.It is also possible to study deformed nuclei, where the deformation is modeled by non-central potentials [21][22][23][24][25]. Recently, the analysis was generalized to consider the nuclear deformations being geometrical.The investigation of this approach motivated the works [26][27][28][29][30] where we solved the Dirac equation in curved space-time.
Following the same idea, we will consider an analogous system in this work, but with the tensor potential with a Coulomb-type term plus a Yukawa-type term, i.e., the Hellmann potential.We will analyze the spin and pseudo-spin symmetries in a deformed nucleus.In this way, our objective is to investigate the behavior of a spin 1/2 particle, with spin and pseudo-spin symmetry, in curved space with this type of interaction through two different ways of coupling the tensor potential.Firstly, we will consider the Dirac equation in curved space-time with the pseudo-centrifugal present and we will analyze the radial spinor and eigenenergies approximately using the Pekeris approximation.In the second analysis, we will consider that the tensor potential nullifies the pseudo-centrifugal potential, thus we get exact analysis of the radial spinor and eigenenergies.So, we will use the Dirac equation with spherical symmetry in a curved space-time where the line element is given by, with f (r ) and g(r ) being arbitrary functions of the radial coordinate r .Equation ( 1) has as special cases Schwarzschild [31], Reissner-Nordström [32] and anti de-Sitter [33] metrics.Because of spherical symmetry of system, the angular Dirac spinor will be given by the spin angular functions, written in terms of spherical harmonics, in the same way as systems with the same symmetry in flat space-time.The paper was organized as follows: in Sect.2, we consider the Dirac Hamiltonian in spherical coordinate in the curved space-time for a general metric and symmetric spherical external electromagnetic field.Then we solve the angular equation to obtain the exact wave function.In Sects. 3 and 4, we solve the radial Dirac-Morse problem with Hellmann tensor potential in curved space-time with spin and pseudo-spin symmetries approximately and exactly, respectively.Finally, in Sect.5, we present our conclusions.

Dirac Equation in Curved Space-Time
From the Eq.(1), the line element will be given by, where α is the fine structure constant and we are considering that e g(r ) = e f (r ) .Considering as EM field given by A μ = (V (r ), c A(r ), 0, 0), the Dirac equation for the coupling of this line element and EM field can be represented in the diagonal gauge or in the cartesian gauge [34].In the diagonal gauge (D) the Dirac equation reads, where α i D = βγ i , β = γ 0 and γ i are the well-known Dirac matrices, for i = 1, 2, 3.In the cartesian gauge (C), the Dirac equation is written as, where α i C = βγ i C , and γ i C are given by, We consider the minimum coupling p 0 → p 0 − A 0 /c and p → p − iβ A/c, as in [35,36], and A = (c A(r ), 0, 0).These two representations are related via similarity transformation S, namely, α i C = Sα i D S −1 and C = S D .The S transformation is defined by, As the system has spherical symmetry, the angular Dirac spinor will be given by the spin angular functions with the Dirac equation in the Cartesian gauge.However, on this meter the Dirac equation is more complicated to work with.With that, we will solve it in the diagonal gauge and then calculate the spinor in the cartesian gauge applying the transformation C = S D .
Considering H = i∂/∂t, the Dirac Hamiltonian in diagonal gauge is given by, From the eigenvalue problem given by H D = E D , where E and D are eigenenergy and eigenfunction, respectively, and applying a unitary transformation given by U = (1 4 + γ 2 γ 1 + γ 1 γ 3 + γ 3 γ 2 )/2 in (7) we obtain, where H/c changes as follows U (H/c) U † and = U D .Using the angular spinor explicitly calculated in [34], the Dirac spinor in cartesian gauge will be, where ≡ C , l is the pseudo-orbital angular momentum given by l = l + 1 and Y |m| j l are the spinorial functions given by where Y m l are the spherical harmonics [37].The radial functions R 1 (r ) and R 2 (r ) must satisfy the system, ⎛ ⎜ ⎜ ⎝ where = E/c 2 .Having determined the angular spinor of the system, we can now solve the radial spinor in the differential equations (11).Taking e f = 1 + α 2 U (r ) we will consider all the information of curvature in space-time at scalar potential U (r ), so using (11) we obtain, where A e f f (r ) = λ r + λα 2 U (r ) r + A(r ), is the effective tensor potential, which contains terms of spin-orbit coupling, curvature of space and EM field, respectively.
To solve the radial wave functions R 1 (r ) and R 2 (r ) we need to define explicitly the functions V (r ), U (r ) and A(r ).In the next sections we will considering the Dirac equation with Morse scalar and vector potential and Hellmann-type tensor potential with spin and pseudo-spin symmetries.In Sect. 3 we will analyze the system with non-zero effective potential and in Sect. 4 with zero effective tensor potential.

Approximate Solution of Dirac-Morse Problem with Spin Symmetry
The first case we are going to analyze is the relativistic Morse-type scalar and vector potential and Hellmanntype tensor potential with spin symmetry.Thus, we have (r r , where A, μ, r e , and δ are real constants.The effective tensor potential will be given by A e f f (r ) = (λ + A)/r .Thus, from ( 16) we obtain, substituting x = (r − r e )/r e into (49), we obtain where β = l(l + 1) + A(2λ + A + 1), with λ(λ + 1) = l(l + 1).As far as we know, Eq. ( 22) does not have an exact solution, so an alternative is to solve it in an approximate way.Let us analyze the system for small oscillations, namely for r ≈ r e , i.e, for x ≈ 0. For this, we will apply Pekeris approximation, which is an expansion in the term centrifuge 1/(1 + x) 2 for x = 0 until second order.Thus and we want to write the term 1/(1 + x) 2 as β 0 + β 1 e −δr e x + β 2 e −2δr e x , thus thus, equalizing ( 23) and ( 24), we obtain β 0 = 1 + 3/(δ 2 r 2 e ) − 3/(δr e ), β 1 = 4/(δr e ) − 6/(δ 2 r 2 e ) and β 2 = 3/(δ 2 r 2 e ) − 1/(δr e ).From ( 22), we have Fig. 1 Plots of some energy spectra for values other than A, where we use μ = 0.01, r e = 2.40873 and δ = 0.988879.We note that for A = 0 there will be a break of degenerescence in the doublets states.We use atomic units The above equation has already been solved in [38] in the context of the Morse-Schrödinger problem.Such equation is given by, and its solution is, with where √ ββ 2 and for the energy to be real we have to impose, It is important to note that with the appearance of the effective tensor potential given by A e f f = (λ + A)/r , this causes the eigenenergies given in (29) to depend on the spin-orbit coupling term given by λ.Thus, that for each choice of λ given in (17) we will have a different eigenenergies value, thus resulting in the break of degenerescence between the doublets states (n, l, l − 1/2) and (n, l, l + 1/2), causing the spin symmetry to be broken.However, when we make A = 0 we eliminate the dependence of the term λ on eigenenergies, that is, we return the spin symmetry with degenerescence, e.g., in the doublet states (1 p 1/2 , 1p 3/2 ), as we can observe in the Fig. 1.
We have that R 1 will be given by,  [37].Finally, R 2 (r ) will be given by, similarly to what we did in 1/r 2 , where we expanded around r e and rewrote as exponential, here we will do the same with the expansion in term 1/r , then replacing it x = (r − r e )/r e , we obtain with ) and k 2 = 1/(δ 2 r 2 e ) − 1/(2δr e ), so we have We obtain spinorial wave function for r ≈ r e given by, where we write The spinorial wave function in (35) can be written as curved = (g tt ) −1/4 f lat where g tt = (1 + α 2 μ exp[−δ(r − r e )]) 2 is the first element of the metric given in (36) and f lat is the spinorial wave function in flat space-time with effective mass m(r The physical interpretation of this system in curved space-time is that with the coupling of the electromagnetic field A μ = (μ exp[−δ(r − r e )], c A/r − αλμ exp[−δ(r − r e )]/r, 0, 0) with the metrics given in (36) we fall into the problem of an effective mass particle m(r ) = 1 + α 2 μ exp[−δ(r − r e )] and effective tensor potential A/r on flat space-time, so we can write the Dirac spinor solution in curved space-time as An important detail is that if we obtained these results for an approximate case (r ≈ r e ), there may have been a loss of generality in the analysis of the symmetry break due to the tensor potential.But, it is interesting to note that regardless of how we expand the term 1/(1 + x) 2 in (23), that is, independent if we expand around x ≈ 0 or no, and we go to second order or higher, what will change will be only the quantity and the values of the parameters that will need to be adjusted, which in our case were called β 0 , β 1 and β 2 .In fact, we see that the contribution of the tensor potential in the eigenenergies is due to the parameter β in the equation (26), with this we conclude that there will be a breaking in the spin symmetry for every domain of r in this system, without loss of generality.
In summary, we obtained the Dirac spinor given in (35) and eigenvalues energies given in (29) of the Morse-type scalar and vector potential and Hellmann-type tensor potential on curved space-time whose metric is The scalar curvature [31] of ( 36) is given by, so we have that for r → 0 we obtain R → ∞, thus the curvature is infinity and we have an singularity.When r → ∞ we obtain R → 0 in this limit, so the space-time will be flat.

Approximate Solution of Dirac-Morse Problem with Pseudo-Spin Symmetry
Let us consider once more the Morse-type scalar and vector potential and Hellmann-type tensor potential, but now with pseudo-spin symmetry.So we have (r ) = 0, (r ) = 2V (r ) = 2μ exp[−δ(r − r e )] and A(r ) = A/r − α 2 λμ exp[−δ(r − r e )]/r .Again, the effective tensor potential is A e f f (r ) = (λ + A)/r .Thus of ( 19) we obtain, substituting x = (r − r e )/r e into the above equation yields, with β = l( l + 1) + 2 Aλ − A + A 2 .Using the same reasoning as in the previous section, we will use the Pekeris approach in the centrifugal term 1/(1 + x) 2 .Thus using ( 25), (39) results in, comparing with (27) we obtain √ ββ 2 and the energies is given by, where γ = −δr e /2 − ββ 1 /(2 √ ββ 2 ) + μr 2 e / √ ββ 2 and for the energy to be real we have to have, We observed, as in the previous case, that an effective tensor potential given by A e f f = (λ + A)/r appears, this makes the eigenenergies given in (41) depend on the spin-orbit coupling given by λ, and since λ has two possible values, for each of these values we will have a different energy value, thus resulting in the break of degenerescence between the double states ( ñ, l, l − 1/2) e ( ñ, l, l + 1/2), causing the pseudo-spin symmetry to be broken spin.However, again when we make A = 0 we eliminate the dependence of the term λ on eigenenergies, that is, it returns the pseudo-spin symmetry with degeneration, for example, in states (1 p1/2 , 1 p3/2 ), as noted in the Fig. 2. Again, there is no loss of generality in the symmetry breaking due to the analysis being made for an approximate case.
We have that R 2 will be given by ( 28), with s = √ ββ 2 e δr e /(δr e ) and ν n = −(δr e /2 + ββ 1 /(2 √ ββ 2 ) − μr 2 e / √ ββ 2 )/(δr e ) − n ≥ 0. Finally, R 1 will be given by, doing the substitution x = (r − r e )/r e and using (33), we obtain Fig. 2 Plots of some energy spectra for values other than A, where we use μ = 0.01, r e = 2.40873 and δ = 0.988879.We note that for A = 0 there will be a break of degenerescence in the doublets states.We use atomic units Finally we obtain spinorial wave function for r ≈ r e , (r, θ, φ) ≈ where we write As in case previous, the spinorial wave function in (35) can be written 2 is the first term of the line element given in (47) and f lat is the spinor Dirac in flat space-time with effective mass m(r Again, here we have that the physical interpretation of this system in curved space-time is that with the coupling of the electromagnetic field A μ = (μ exp[−δ(r − r e )], c A/r − αλμ exp[−δ(r − r e )]/r, 0, 0) with the metric given in (47) we fall into the problem of an effective mass particle m(r ) = 1 − α 2 μ exp[−δ(r − r e )] and effective tensor potential A/r on flat space-time, so that we can write Dirac's spinor solution in curved space-time as curved = (1 − α 2 μ exp[−δ(r − r e )]) −1/2 f lat .
In summary, we obtained the spinor Dirac given in (46) and eigenvalues energies given in (41) of the Morse-type scalar and vector potential and Hellmann-type tensor potential on curved space-time whose metric is The scalar curvature [31] of (47) is given by, so we have that for r → 0 we obtain R → ∞, thus the curvature is infinity and we have an singularity.When r → ∞ we obtain R → 0, so the space-time will be flat in this limit.

Dirac-Morse Problem Without Pseudo-Centrifugal Potential
In this second analysis, we will consider the tensor potential given by A(r ) = −λ/r − α 2 λU (r )/r , with ] is the generalized Morse potential.Thus, we have a potential similar to the previous one, but now this potential is quantized because it is dependent on λ.In this way we will have A e f f (r ) = 0, such that we will eliminate the dependence of the pseudo-centrifugal potential in the calculation of the radial spinor.In this case we will use r e = 0 for the sake of convenience, as we will not make approximations.

Exact Solution of Dirac-Morse Problem with Spin Symmetry
For this first case with spin symmetry we have (r and δ are real constants.Thus, from ( 16) we obtain, Again, the above equation has already been solved in [38] in the context of the Morse-Schrödinger problem, thus comparing with the Eq. ( 27), we obtain and we obtain, when mapping with the non-relativistic case, we verify that the coupling constants V 1 and V 2 can be written as a function of the eigenergies.The eigenenergies is given by, and for the energy to be real we have to impose, it is important to note that by fact of effective tensor potential be A e f f = 0, this causes the eigenenergies given in (52) do not depend on the spin-orbit coupling constant λ.Thus, unlike the analysis in Sect.3, the eigenergies will not depend on the spin quantum number, even though there is a tensor potential A(r ) coupled to the system, thus the spin symmetry is maintained, as there is degeneracy between the doublets states (n, l, l − 1/2) and (n, l, l + 1/2).Furthermore, we also observe that there will be degeneracy with respect to the orbital angular momentum, since the eigenenergies also do not depend on l.Finally, R 2 (r ) will be given by 15, and we obtain spinorial wave function given by, where we write The spinorial wave function in (55) can be written as curved = (g tt ) −1/4 f lat where 2 is the first element of the metric given in (36) and f lat is the spinorial wave function in flat space-time with effective mass m(r The physical interpretation of this system in curved space-time is that with the coupling of the electromagnetic field )/r, 0, 0 with the metrics given in (56) we fall into the problem in flat space-time of an effective mass particle m(r and effective tensor potential null, so we can write the Dirac spinor solution in curved space-time as In the Figs. 3 and 4, the normalized probability densities |ψ c (r ) ) are analyzed for different values of the coupling constants V 1 and V 2 through A and B, in order to observe how the coupling of the field A μ with the metric in (56) influences the behavior of the particle.We note that the greater the value of B, the closer to the origin in r the particle is confined, where its mass is smaller-m(r ) = 1 + α 2 (V 1 e −2δr − V 2 e −δr ).We use atomic units Fig. 4 Plots of some normalized probabilities density for values different of A, where we use n = 0, B = 0.1, δ = 0.1 and the positive eigenenergies.Again, we observe that the greater the value of A, furthest from the origin in r the particle is confined, where its mass is smaller-m(r ) = 1 + α 2 (V 1 e −2δr − V 2 e −δr ).We use atomic units From this figures, we have that the probability density is confined in different regions according to the variation we make in the Morse potential.Thus, we observe in the Fig. 3 that by fixing A and varying the value of B, the particle is confined in regions closer and closer to the origin in r the greater the value of B, where its mass is less-m(r ) = 1 + α 2 (V 1 e −2δr − V 2 e −δr ).On the other hand, we observe the Fig. 4, fixing the constant B and varying the value of A, we see that the particle will also be confined in regions where the mass is smaller the greater the value of A, but now in regions farther from the origin in r .
In summary, we obtained exactly the Dirac spinor given in (35) and eigenvalues energies given in (29) of the Dirac-Morse problem with spin symmetry in the presence of Hellmann-type tensor potential in curved space-time whose metric is The scalar curvature [31] of ( 36) is given by, where U (r ) = V 1 e −2δr − V 2 e −δr , so we have that for r → 0 we obtain R → ∞, thus the curvature is infinity and we have an singularity.When r → ∞ we obtain R → 0 in this limit, so the space-time will be flat.

Exact Solution of Dirac-Morse Problem with Pseudo-Spin Symmetry
For this case with pseudo-spin symmetry we have (r ) = 0, (r and δ are real constants.Thus, from (19) we obtain, Again, the above equation has already been solved in [38] in the context of the Morse-Schrödinger problem, thus comparing with the Eq. ( 27), we obtain with μ = 2 A/δ and ν ñ = B/δ− ñ.So in our case we have θ = δ, and we obtain, when mapping with the non-relativistic case, we verify, again, that the coupling constants V 1 and V 2 can be written as a function of the eigenenergies.The eigenenergies is given by, and we note that is same of previous case, and for the energy to be real we have to impose, Once again the effective tensor potential A e f f = 0, this causes the eigenenergies given in (61) do not depend on the spin-orbit coupling constant λ.Thus, unlike the analysis in Sect.3, the eigenenergies will not depend on the spin quantum number, even though there is a tensor potential A(r ) coupled to the system, thus the pseudo-spin symmetry is maintained, as there is degeneracy between the doublets states ( ñ, l, l − 1/2) and ( ñ, l, l + 1/2).Furthermore, there will be degeneracy with respect to the orbital angular momentum, since the eigenenergies also do not depend on l.
Finally, R 1 (r ) will be given by (18), and we obtain spinorial wave function given by, where we write R 1 = R 1 ñ l and R 2 = R 2 ñ l .The spinorial wave function in (64) can be written as curved = (g tt ) −1/4 f lat where g 2 is the first element of the metric given in (36) and f lat is the spinorial wave function in flat space-time with effective mass m(r The physical interpretation of this system in curved space-time is that with the coupling of the electromagnetic field )/r, 0, 0 with the metrics given in (65) we fall into the problem in flat space-time of an effective mass particle m and effective tensor potential null, so we can write the Dirac spinor solution in curved space-time as In the Figs. 5 and 6, the normalized probability densities |ψ c (r ) are analyzed for different values of the coupling constants V 1 and V 2 through A and B, in order to observe how the coupling of the field A μ with the metric in (65) influences the behavior of the particle.
Again, as previous case, we have that the probability density is confined in different regions according to the variation we make in the Morse potential.Thus, we observe in the Fig. 5 that by fixing A and varying the value of B, the particle is confined in regions closer and closer to the origin in r the greater the value of B, where its mass is less-m(r ) = 1 − α 2 (V 1 e −2δr − V 2 e −δr ).On the other hand, we observe the Fig. 6, fixing the constant B and varying the value of A, we see that the particle will also be confined in regions where the mass is smaller the greater the value of A, but now in regions farther from the origin in r .5 Plots of some normalized probabilities density for values different of B, where we use ñ = 0, A = 0.1, δ = 0.01 and the negative eigenenergies.We note, once again, that the greater the value of B, the closer to the origin in r the particle is confined, where its mass is smaller-m(r ) = 1 − α 2 (V 1 e −2δr − V 2 e −δr ).We use atomic units Fig. 6 Plots of some normalized probabilities density for values different of A, where we use n = 0, B = 0.1, δ = 0.1 and the negative eigenenergies.Again, we observe that the greater the value of A, furthest from the origin in r the particle is confined, where its mass is smaller-m(r ) = 1 − α 2 (V 1 e −2δr − V 2 e −δr ).We use atomic units In summary, we obtained exactly the Dirac spinor given in (46) and eigenvalues energies given in (41) of the Dirac-Morse problem with pseudo-spin symmetry in the presence of Hellmann-type tensor potential in curved space-time whose metric is The scalar curvature [31] of (65) is given by, where U (r ) = V 1 e −2δr − V 2 e −δr , so we have that for r → 0 we obtain R → ∞, thus the curvature is infinity and we have an singularity.When r → ∞ we obtain R → 0 in this limit, so the space-time will be flat.We note that in this current work the spin and pseudo-spin symmetry will be broken due to the effective tensor potential coupled to the system when we analyze of approximate form in Sect.3, as well as in the analyzes of the harmonic oscillator [26], coulomb potential [27] and even kratzer potential [28], all cases in curved space-time.In all these cited cases, the spin and pseudo-spin symmetry is recovered by adjusting the value of the coupled potentials, making the effective tensor potential zero.However, when we analyze the problem exactly by eliminating the pseudo-centrifugal term, we can consequently eliminate the dependence of the spin-orbit coupling on the eigenenergies, since we consider the tensor potential dependent on the spin-orbit coupling term-λ/r , so the coupling with the space curvature maintains the degeneracy in the eigenenergies with respect to l and j, consequently the spin and pseudo-spin symmetries.That is, the essential difference of the last case for the previous ones and the approximate one made in this work, is that in them there is dependence on the spin-orbit coupling term in the coupling, since an effective tensor potential arises through the electromagnetic field with the line element in the curved space-time, so the symmetry is broken.

Fig. 3
Fig.3Plots of some normalized probabilities density for values different of B, where we use n = 0, A = 1, δ = 0.5 and the positive eigenenergie.We note that the greater the value of B, the closer to the origin in r the particle is confined, where its mass is smaller-m(r ) = 1 + α 2 (V 1 e −2δr − V 2 e −δr ).We use atomic units