In this work, the generalized Dirac oscillator in cosmic string space-time is studied by replacing the momentum pμ with its alternative pμ+mωβfμxμ. In particular, the quantum dynamics is considered for the function fμxμ to be taken as Cornell potential, exponential-type potential, and singular potential. For Cornell potential and exponential-type potential, the corresponding radial equations can be mapped into the confluent hypergeometric equation and hypergeometric equation separately. The corresponding eigenfunctions can be represented as confluent hypergeometric function and hypergeometric function. The equations satisfied by the exact energy spectrum have been found. For singular potential, the wave function and energy eigenvalue are given exactly by power series method.

National Natural Science Foundation of China115650091. Introduction

In quantum mechanics, there has been an increasing interest in finding the analytical solutions that play an important role for getting complete information about quantum mechanical systems [1–3]. The Dirac oscillator proposed in [4] is one of the important issues in this relativistic quantum mechanics recently. In this quantum model, the coupling proposed is introduced in such a way that the Dirac equation remains linear in both spatial coordinates and momenta and recovers the Schrödinger equation for a harmonic oscillator in the nonrelativistic limit of the Dirac equation [4–11]. As a solvable model of relativistic quantum mechanical system, the Dirac oscillator has many applications and has been studied extensively in different field such as high-energy physics [12–15], condensed matter physics [16–18], quantum Optics [19–25], and mathematical physics [26–33]. On the other hand, the analysis of gravitational interactions with a quantum mechanical system has recently attracted a great deal attention and has been an active field of research [5, 6, 34–43]. The study of quantum mechanical problems in curved space-time can be considered as a new kind of interaction between quantum matter and gravitation in the microparticle world. In recent years, the Dirac oscillator embedded in a cosmic string background has inspired a great deal of research such as the dynamics of Dirac oscillator in the space-time of cosmic string [44–47], Aharonov-Casher effect on the Dirac oscillator [5, 48], and noninertial effects on the Dirac oscillator in the cosmic string space-time [49–51]. It is worth mentioning that based on the coupling corresponding to the Dirac oscillator a new coupling into Dirac equation first has been proposed by Bakke et al. [52] and used in different fields [53–57]. This model is called the generalized Dirac oscillator which in special case is reduced to ordinary Dirac oscillator. Inspired by the above work, the main aim of this paper is to analyze the generalized Dirac oscillator model with the interaction functions fμxμ taken as Cornell potential, singular potential and exponential-type potential in cosmic string space-time and to find the corresponding energy spectrum and wave functions. This work is organized as follows. In Section 2, the new coupling is introduced in such a way that the Dirac equation remains linear in momenta, but not in spatial coordinates in a curved space-time. In Sections 3, 4, and 5, we concentrate our efforts in analytically solving the quantum systems with different function fμxμ and find the corresponding energy spectrum and spinors, respectively. In Section 6, we make a short conclusion.

2. Generalized Dirac Oscillator with a Topological Defect

In cosmic string space-time, the general form of the cosmic string metric in cylindrical coordinates read [41, 42, 44, 58, 59](1)ds2=-dt2+dρ2+α2ρ2dφ2+dz2,with -∞<t,z<+∞, 0<ρ<+∞, and 0<φ<2π. The parameter α is related to the linear mass density of string η by α=1-4η and runs in the interval 0,1. In the limit as α→1 we get the line element of cylindrical coordinates. The Dirac equation in the curved space-time ħ=c=1 reads(2)iγμx∂μ-iγμxΓμx-mψt,x=0,where the γμ matrices are the generalized Dirac matrices defining the covariant Clifford algebra γμ,γν=2gμν, m is mass of the particle, and Γμ is the spinor affine connection. We choose the basis tetrad eaμ as(3)eaμ=10000cosφsinφ00sinφαρcosφαρ00001,then in this representation the matrices γμ [44] can be found to be(4)γ0=γt,γ1=γρ=γ1cosφ+γ2sinφ,γ2=γφ=-γ1sinφ+γ2cosφ,γ3=γz,γμΓμx=1-α2αργρ.It is well known that, in both Minkowski space-time and curved space-time, usual Dirac oscillator can be obtained by the carrying out nonminimal substitution pμ⟶pμ+mωβxμ in Dirac equation where m and ω are the mass and oscillator frequency. In the following, we will construct the generalized oscillator in curved space-time. To do this end, we can replace momenta pμ in the Dirac equation of curved space-time by(5)pμ⟶pμ+mωβfμxμ,where fμxμ are undetermined functions of xμ. It is to say that we introduce a new coupling in such a way that the Dirac equation remains linear in momenta but not in coordinates. In particular, in this work, we only consider the radial component the nonminimal substitution(6)fμxμ=0,fρρ,0,0.By introducing this new coupling (6) into (2) and with the help of (4), in cosmic string space-time the eigenvalue equation of generalized Dirac oscillator can be written as(7)-iγt∂t+iγρ∂ρ-1-α2αρ+mωρfρ+iγφ∂φαρ+iγz∂z-mψ=0.We choose the following ansatz:(8)ψ=e-iEt+il+1/2-Σ3/2φ+ikzχ1ρχ2ρ,then we have(9)α1ddρ+12ρ-mωρfρ-λρα2-kα3α1ddρ+12ρ+mωfρ-λρα2-kα3χ1=E2-m2χ1,(10)α1ddρ+12ρ+mωfρ-λρα2-kα3α1ddρ+12ρ-mωfρ-λρα2-kα3χ2=E2-m2χ2,where(11)α1=iσ1cosφ+σ2sinφ,α2=-σ1sinφ+σ2cosφ,α3=σ3.It is straightforward to prove the following relations satisfied above matrices αi:(12)α12=-α22=α32=-1,α1α2=-α2α1=iσ1σ2,α1α3=-α3α1=iσ1σ3cosφ+σ2σ3sinφ,α3=-α3α2=-σ1σ3sinφ+σ2σ3cosφ.With help of (12) and simple algebraic calculus, (9) becomes(13)∂ρ2+1ρ∂ρ-1ρ214+iλσ1σ2+λ2χ1+-2mωfρρikρσ1σ3cosφ+σ2σ3sinφ+iσ1σ2λχ1+m2+k2-E2+mωfρρ-m2ω2f2ρχ1=0.It is easy to prove the following relation [44]:(14)iσ1σ2λ+ikρσ1σ3cosφ+σ2σ3sinφ=-2s→.L→,where s→=σ→/2. The eigenvalue of s→.L→ can be assumed as l+1/2/2α and (13) reads∗ad2χ1dρ2+1ρddρχ1-λ2ρ2+μfρρ-mωdfρdρ+m2ω2f2ρχ1+νχ1=0,where(15)λ=l+1/2α-12,μ=-2l+1/2mωα,ν=E2-m2-k2.For the component χ2, from (10) an analog equation can be also obtained∗bd2χ2dρ2+1ρddρχ2-λ2ρ2+μfρρ+mωdfρdρ+m2ω2f2ρχ2+νχ2=0,where(16)λ=l+1/2α+12,μ=-2l+1/2mωα,ν=E2-m2-k2.In particular, ∗a and ∗b will be reduced to the result obtained in [44] when the function fρ is taken as fρ=ρ. As we can see, ∗a and ∗b have the same form. So without loss of generality in remaining parts of this work, our main tasks is only to solve the equation ∗a with different functions fρ and find corresponding eigenvalue and eigenfunction. While with regard to ∗b, it is straightforward to obtain the corresponding solution.

3. The Solution with <inline-formula>
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<bold>(<italic>ρ</italic>)</bold> to Be Cornell Potential

The Cornell potential that consists of Coulomb potential and linear potential has gotten a great deal of attention in particle physics and was used with considerable success in models describing systems of bound heavy quarks [60–62]. In Cornell potential, the short-distance Coulombic interaction arises from the one-gluon exchange between the quark and its antiquark, and the long-distance interaction is included to take into account confining phenomena.

Now we let the function fρ be Cornell potential(17)fρ=aρ-bρ,where a and b are two constants. Substituting (17) into ∗a and ∗b leads to following equation:(18)d2χdρ2+1ρddρχ+-τ12ρ2-τ2ρ2+τ3χ=0,where(19)τ12=λ2-μb+m2ω2b2-ωmb,τ2=m2ω2a2,τ3=υ+2abm2ω2-aμ+mωa.We make a change in variables ξ=mωaρ2 and then (18) can be rewritten as(20)ξd2χdξ2+ddξχ+-τ124ξ-14ξ+τ34maωχ=0.Taking account of the boundary conditions satisfied by the wave function χ, i.e., χ∝ξτ1/2 for ξ→0 and χ∝e-ξ/2 for ξ→∞, physical solutions χ can be expressed as [44, 60, 63](21)χ=ξτ1/2e-ξ/2gξ.If we insert this wave function χ into (20), we have the second-order homogeneous linear differential equation in the following form:(22)ξd2gdξ2+τ1+1-ξddξg+τ34maω-τ12-12g=0.It is well known that (22) is the confluent hypergeometric equation and it is immediate to obtain the corresponding eigenvalues and eigenfunctions(23)gξ=F-τ34maω-τ1+12,τ1+1,ξ,(24)En2=δ1-2l+1/2mωaα+4maωn+12+δ22,with(25)δ1=m2+k2-2abm2ω2+mωa,δ2=λ2-μb+b2m2ω2-mωb.In particular, if we assume that α=1, from (24), the energy levels of generalized Dirac oscillator with fρ to be Cornell potential in the absence of a topological defect can be obtained. In addition if we let a=1, b=0 in (24) the energy levels given here will be reduced to that one obtained in [44].

4. The Solution with <inline-formula>
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<bold>(<italic>ρ</italic>)</bold> to Be Singular Potential

The investigation of singular potentials in quantum mechanics is almost as old as quantum mechanics itself and covers a wide range of physical and mathematical interest because the real world interactions were likely to be highly singular [64]. The singular potentials of vr∝1/rn type, with n≥2, are of great current physical interest and are relevant to many problems such as the three-body problem in nuclear physics [65, 66], point-dipole interactions in molecular physics [67], the tensor force between nucleons [68], and the interaction between a charges and an induced dipole [69], respectively. Recently, in cosmic string background, singular inverse-square potential with a minimal length had been studied [70].

Next let us take fρ to be singular inverse-square-type potential [71](26)fρρ=a+bρ+cρ2.Substituting (26) into ∗a, the corresponding radial equation reads(27)d2dρ2+1ρddρ-δ1ρ-λ2+δ2ρ2-δ3ρ3-δ4ρ4+γχ=0,where (28)γ=υ-m2a2ω2δ1=2abm2ω2+aμ,δ2=μb+mωb+b2m2ω2+2acm2ω2,δ3=μc+2mωc+2bcm2ω2,δ4=c2m2ω2.It is obvious that (27) has the same mathematical structure with the Schrödinger equation of fourth-order inverse-potential in [72]. So (27) can be solved by power series method.

We look for an exact solution of (27) via the following ansatz to the radial wave function [72–74]:(29)χ=Θρexpgρ,gρ=-δ12ρ-δ22ρ-δ32logρ.Thence, (27) can be transformed into the following form:(30)d2dρ2+-δ1+1-δ3ρ+δ2ρ2ddρ+γ+δ124+δ1δ3-32ρ+δ32-4δ2-2δ1δ2-4λ24ρ2+-δ21+δ3-δ32ρ3+δ22-4δ44ρ4Θρ=0.Now we take Θρ in following series form:(31)Θρ=∑n=0∞anρn+λ+1/2,a0≠0,a1≠0.Substituting (31) into (30) gives rise to following equation:(32)∑n=0∞an-δ1n+λ+2-δ32ρn+λ-1/2+γ+δ124ρn+λ+1/2+2n+2λ+12n+2λ+1-2δ34-δ2δ1+22-λ2+δ324ρn+λ-3/2-δ1n+λ+2-δ32ρn+λ-1/2+γ+δ124ρn+λ+1/2=0.To make (32) be valid for all values of ρ, the coefficients of each term of the polynomial of ρ must be equal to zero separately. We, therefore, obtain(33a)2δ1+2n+λ+2=4λ2-9-4,(33b)δ2=-n+λ+2,(33c)δ3=2n+λ+2,(33d)2δ4=n+λ+22,(33e)γ=-δ124.Using ∗a, (33a), and (33e) and after simple algebraic calculation, the corresponding energy can be written as(34)En2=m2+k2+m2a2ω2-1164λ2-9n+λ+22.The general radial wave functions corresponding to the energy spectra given in (34) are(35)Θρ=∑n=0∞anρn+λ+1/2exp-δ12ρ-δ22ρ-δ32logρ.With the help of (32) and (33a)–(33e), the expansion coefficients an in (35) satisfy the following recursion relation [72]:(36)an+1=4λ2-9-4n+λ+22n+λ+22an-1.From the recursion relation (36) we can determine the coefficients ann≠0,1 of the power series in terms of a0 and a1. In addition the above recursion relation implies that (35) yields one solution as a power series in even powers of ρ and another in odd powers of ρ.

In addition, (27) can be also mapped to the double-confluent Heun equation by appropriate function transformation [75]. So when fρ is taken as singular inverse-square-type potential, the solutions of (27) can be also given by the solution of the double-confluent Heun equation [75, 76].

5. The Solution with <inline-formula>
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<bold>(<italic>ρ</italic>)</bold> to Be Exponential-Type Potential

The exponential-type potentials are very important in the study of various physical systems, particularly for modeling diatomic molecules. The typical exponential-type potentials include Eckart potentials [77], the Morse potential [78, 79], the Wood–Saxon potential [80], and Hulthén potential [81, 82]. The research work on the Dirac equation with the above potential is mainly concentrated on Minkowski time and space. However, it has been noticed recently that it is also interesting to study this kind quantum systems in a cosmic string background [83]. In this section we will take the fρ as exponential-type function and solve the corresponding Dirac equation in cosmic string space-time.

As is known to all, the Dirac equation and Schrödinger equation have been studied by resorting different methods. A usual way is transforming the eigenvalue equation of quantum system considered into a solvable equation via suitable variable substitutions and function transformations [84–86]. In order to obtain solution for fρ being exponential-type potential, we firstly consider the following linear second-order differential equation(37)x21-x2d2ydx2+x1-x2dydx-L1+L2x-L3x2y=0,where Li,i=1,2,3 are constants. It is known that singular points of a differential equation determine the form of solutions. In this equation, there are two singular points, i.e., x = 0 and x = 1. In order to remove these singularities and get physically acceptable solutions we use the following ansatz:(38)y=xΩ1-xΛRx.where Λ and Ω are two real parameters. Further we make this two parameters to satisfy following relationships:(39)Ω=±L1,Λ=121±1-4L3-L2-L1,and by substituting (38) into (37), the differential equation for χx can be written as(40)1-xx∂2∂x2Rx+2Ω+1-2Ω+2Λ+1x∂∂xRx-Ω+Λ+ΔΩ+Λ-ΔRx=0,with Δ=±-L3. In other words, (37) is reduced to the well-known hypergeometric equation, when condition (39) is imposed. Making use of the boundary conditions at r=0 and r=∞ [86, 87], we can find the equation obeyed by the energy eigenvalue:(41)Ω+Λ+Δ=-n,and the corresponding eigenfunctions is given in terms of the Gauss hypergeometric functions below(42)Rx=AFτ1,τ2;1+2Ω;x,τ1=Ω+Λ+Δ,τ2=Ω+Λ-Δ,where A is normalization constant. Next, we will use the results given here to obtain the solutions of Dirac equation exponential-type interaction in cosmic string space-time. As a direct application of the above method, let us take the function fρ to be as Yukawa potential, Hulthén-Type potential, and generalized Morse potential, respectively.

Case 1 (<inline-formula>
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</inline-formula> being Yukawa potential).

In Yukawa meson theory, the Yukawa potential firstly was introduced to describe the interactions between nucleons [88]. Afterwards, it has been applied to many different areas of physics such as high-energy physics [89, 90], molecular physics [91], and plasma physics [92]. In recent years, the considerable efforts have also been made to study the bound state solutions by using different methods.

Now let us choose fρ to be Yukawa potential(43)fρ=aρe-βρ,then ∗a takes the form(44)d2χdρ2+1ρdχdρ-λ2ρ2+mωaρ2e-βρ+μa+amωβρe-βρ+a2m2ω2ρ2e-2βρχ+νχ=0.However, the radial equation (44) cannot accept exact solution due to the presence of the centrifugal term [85]. In order to find analytical solution, we have to use some approximation approaches for the centrifugal term potential. Following [86], the approximation for the centrifugal term reads(45)1ρ2≈β21-e-βρ2,1ρ≈β1-e-βρ.It is worth mentioning that the above approximations are valid for βρ≪1 [86]. So if we make the control parameter β small enough, then we can guarantee that the above approximations in (45) hold for larger values ρ. In other words, this approximation (45) is valid in our work.

Using the approximation in (45) and setting(46)χ=1ρΘρ,x=e-βρ,in the new function χ and new variable x, (44) becomes(47)x21-x2d2Θdx2+x1-x2dΘdx-L1+L2x-L3x2Θ=0,where(48)L1=-λ2β2+m2+k2-E2,L2=-mωaβ2-aμβ+2m2+2k2-2E2,L3=m2ω2a2-aμβ-amωβ2-m2-k2+E2.Comparing (47) with (37) and using the results given in (41) and (42), it is not difficult to find the equation obeyed by eigenvalues and eigenfunctions and they can be given, respectively,(49)En2-q1-q2-En2+n+121+1+q3-16En2=0,Θρ=Ae-βΩρ1-e-βρΛFτ1,τ2;1+2Ω;e-βρ,where(50)q1=λ2β2-m2-k2,q2=m2+k2+aμβ+amωβ2-m2ω2a2,q3=44m2+4k2-am2ω2,τ1=Ω+Λ+Δ,τ2=Ω+Λ-Δ.

Case 2 (<inline-formula>
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</inline-formula> being Hulthén-type potential).

In this section, we are interested in considering the Hulthén potential that describes the interaction between two atoms and has been used in different areas of physics and attracted a great of interest for some decades [81, 82, 93]. Next we take the interaction function fρ being Hulthén-Type potential(51)fρ=a+be-βρ1-e-βρ,where a, b, and β are real constants. Inserting (45) and (51) into ∗a, then ∗a can written as(52)d2χdρ2+1ρdχdρ-λ2ρ2+μaρ+a2m2ω2+μbρ+2abm2ω2e-βρ1-e-βρχ+b2m2ω2e-βρ+mωbβe-βρ1-e-βρ2χ+νχ=0.In the same way as in previous section, taking into consideration approximation (45) for the centrifugal term and using the variable transformation x=e-βρ and function transformation χ=1/ρΘρ, (51) changes(53)x21-x2d2Θdx2+x1-x2dΘdx-L1+L2x-L3x2Θ=0,where(54)L1=λ2β2+m2ω2a2+βμa+m2+k2-E2,L2=mωbβ+b-aμβ-2mω+2m2+2k2-2E2,L3=-m2ω2a-b2-m2-k2+E2.With the help of (38), (41), and (42), the solutions for fρ being Hulthén-Type potential can be easily obtained and the corresponding eigenvalues and eigenfunctions are, respectively,(55)En2-q4-q5-En2+n+121+1+q6-16En2=0,(56)Θρ=Ae-βΩρ1-e-βρΛFτ1,τ2;1+2Ω;e-βρ,where(57)q4=λ2β2+m2ω2a2+βμa+m2+k2,q5=m2ω2a-b2+m2+k2,q6=4m2+k2+4m2ω22a2-2ab+b2+4β2λ2+4βμa+4mωbβ+4a-bμβ-2mω,τ1=Ω+Λ+Δ,τ2=Ω+Λ-Δ.

Case 3 (<inline-formula>
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</inline-formula> being generalized Morse potential).

The Morse potential [78, 79] as an important molecular potential describes the interaction between two atoms. We choose the interaction function fρ being generalized Morse potential(58)fρ=aρ2e-βρ-e-2βρ.As before, substitution of form (59) into ∗a and straightforward calculation lead to the following equation:(59)d2χdρ2+1ρdχdρ-λ2ρ2+μa+2mωaμρ3e-βρ-e-2βρ-aβmωρ22e-2βρ-e-βρχ+a2m2ω2ρ4e-βρ-e-2βρ2χ+νχ=0.Letting x=e-βρ and χ=1/ρΘρ, the above differential equation (59) changes into the form(60)x21-x2d2Θdx2+x1-x2dΘdx-L1+L2x-L3x2Θ=0,where(61)L1=λ2β2+m2+k2-E2,L2=μa1+2mω+2amωβ-2m2-2k2+2E2,L3=E2-m2ω2a2+2mωaβ-m2-k2.It is easy to see that the differential equation (60) is also similar to (37). So again according to the quantization condition (40) the corresponding expression of eigenvalues can be written as(62)En2-q7-q8-En2+n+121+1+q9-16En2=0,q7=λ2β2+m2+k2,q8=m2ω2a2-2mωaβ+m2+k2,q9=4μa1-2mω-2mωaβ+a2m2ω2+λ2β2.The wave function in this case read(63)Θρ=Ae-βΩρ1-e-βρΛFτ1,τ2;1+2Ω;e-βρ,where(64)τ1=Ω+Λ+Δ,τ2=Ω+Λ-Δ.The above results show that the radial equation of the generalized Dirac oscillator with interaction function fμxμ to be taken as the exponential-type potential can be mapped into the well-known hypergeometric equation and the analytical solutions can have been found.

6. Conclusion

In this work, the generalized Dirac oscillator has been studied in the presence of the gravitational fields of a cosmic string. The corresponding radial equation of generalized Dirac oscillator is obtained. In our generalized Dirac oscillator model, we take the interaction function fμxμ to be as Cornell potential, Yukawa potential, generalized Morse potential, Hulthén-Type potential, and singular potential, respectively. By solving the corresponding wave equations the corresponding energy eigenvalues and the wave functions have been obtained and we have showed how the cosmic string leads to modifications in the spectrum and wave function. Based on consideration that Dirac oscillator has been studied extensively in high-energy physics, condensed matter physics, quantum optics, mathematical physics, and even connection with Higgs symmetry it also makes sense to generalize the generalized Dirac oscillator to these fields.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11565009).

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