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Yang Yi, Cai Shao-Hong, Long Zheng-Wen, Chen Hao, Long Chao-Yun. Exact solution of the (1+2)-dimensional generalized Kemmer oscillator in the cosmic string background with the magnetic field. Chinese Physics B, 2020, 29(7): 070302  
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Exact solution of the (1+2)-dimensional generalized Kemmer oscillator in the cosmic string background with the magnetic field
Yang Yi1, Cai Shao-Hong2, †, Long Zheng-Wen1, ‡, Chen Hao1, Long Chao-Yun1
College of Physics, Guizhou University, Guiyang 550025, China
School of Information, Guizhou University of Finance and Economics, Guiyang 550025, China

 

† Corresponding author. E-mail: caish@mail.gufe.edu.cn zwlong@gzu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11465006 and 11565009).

Abstract

We study a two-dimensional generalized Kemmer oscillator in the cosmic string spacetime with the magnetic field to better understand the contribution from gravitational field caused by topology defects, and present the exact solutions to the generalized Kemmer equation in the cosmic string with the Morse potential and Coulomb-liked potential through using the Nikiforov–Uvarov (NU) method and biconfluent Heun equation method, respectively. Our results give the topological defect’s correction for the wave function, energy spectrum and motion equation, and show that the energy levels of the generalized Kemmer oscillator rely on the angular deficit α connected with the linear mass density m of the cosmic string and characterized the metric’s structure in the cosmic string spacetime.

PACS: ;03.65.Pm;;11.27.+d;;03.65.Ge;
Keyword:;generalized Kemmer oscillator;;cosmic string;;exact solutions;
1. Introduction

The Dirac equation with the linear interactions is known as the Dirac oscillator[1,2] and the interaction of Dirac oscillators can be explained as the interaction between linear electric fields and abnormal magnetic moments.[3,4] There are very few quantum systems that can be accurately solved in quantum mechanics, and the Dirac oscillator (DO) is one of them.[5] The reason why Dirac oscillators have attracted much attention is not only because it is solvable[6–9] but also because it has many physical applications, such as in condensed matter physics,[10,11] high energy physics[12,13] and quantum optics.[14,15] Furthermore, in relativistic quantum mechanics the Kemmer equation is Dirac-liked relativistic wave equation,[16] which was first proposed by Kmmer in 1939. Bednar et al. consider that the massive spin-1 particles form a two-particle system with spin 1/2, rather than a single particle with spin 1. As a consequence, the Kemmer equation is a two-body Dirac-liked equation.[17] Recently, this equation has particularly obtained a lot of interests. It has been investigated under the background of five-dimensional Galilean invariance,[18] in the context of quantum chromodynamics[19] and in the existence of some form interactions.[20,21]

The wave function’s exact solutions for relativistic quantum mechanics are quite significant to comprehending the physics meaning about such solutions. They are a meritorious tool for determining the radiative contribution for energy. In many different cases, different techniques have been used to study the quantum mechanics of spin-1, massed and charged particles in the external field. These works specifically studied the solution to wave equation in a magnetic field.[22–24] There are a host of ways that can be used to study the solution to the Kemmer equation, such as the functional Bethe ansatz method,[25,26] the supersymmetric quantum mechanics method[27,28] and the Nikiforov–Uvarov method.[29,30] Particularly, the Nikiforov–Uvarov method has been used extensively to solve many quasi-accurate models in theoretical physics research.[31–33]

Recent years, the quantum systems in curved spacetime background has motivated a great number of investigations.[34–39] The cosmic string, which is a linear defect, will transform the medium’s topology as globally observed. The general relativity deems that gravitation is a curvature of spacetime and the Riemann tensor is used to describe the curvature. As is well-known, due to the fact that the atom interact with curvature spacetime, the atom’s energy levels in gravitational field can be transformed. Hence, for the sake of completely describing the physical system, the topology of the spacetime should be considered. The impact of topology for the two-dimensional Dirac oscillator, the Dirac field and the Dirac particle restricted in the Aharonov–Bohm ring has been studied.[40–42] In Ref. [43] the authors have studied the one-dimensional generalized Kemmer oscillator in the flat spacetime. In Ref. [44], the authors have studied the two-dimensional Kemmer oscillator in the curve spacetime. In this work, we extend the two-dimensional Kemmer oscillator to the two-dimensional generalized Kemmer oscillator in the curved spacetime to explore the topology defects’ effects on the generalized Kemmer oscillator.

In this work, the relativistic quantum dynamics for the generalized Kemmer oscillator in the cosmic string background with the magnetic field is studied. According to the corresponding Kemmer equation, the topological defects on the motion equation, the wave function and the energy spectrum are analyzed. The paper is organized as follows. In Section 2, we present the generalized two-dimensional Kemmer oscillator in cosmic string background with the magnetic field. In Section 3, we devote the precise solutions to the generalized Kemmer equation under the Morse potential, using the Nikiforov–Uvarov method, and Coulomb-liked potential, using the biconfluent Heun equation method. Finally, the conclusion is given in Section 4.

2. The generalized Kemmer oscillator in the cosmic string background with the magnetic field

The line element, in cylindrical coordinates, of cosmic string spacetime can be written as[45–47]

where –∞ < (t, z) > + ∞, r ≥ 0 and 0 ≤ ϕ ≤ 2 π. The angular deficit parameter α has the ranges (0, 1], which has the following relation with linear mass density (m) of the cosmic string: α = 1 – 4 m. The Kemmer equation in curved spacetime can be written as[44]

where covariant derivative is ∇μ = ∂μ – Σμ, ,[48] Γμ are the spinor affine connections, and is the 4 × 4 identity matrix. Moreover, , M and ⊗ denote the Kemmer matrices decided by the line element, the mass of spin-1/2 particles and direct product, respectively. The Kemmer matrices under the cosmic string background satisfy the commutation relation[44]

with . Here γμ(x) are the generalized Dirac matrices that satiate the relations[49]

and it is defined on the basis of a set of tetrad . Meanwhile, satisfies the relation , γa represents the Dirac matrices in the flat spacetime, and the tensor ηab is the Minkowski spacetime metric tensor. The spinorial affine connection is given by[48]

The Dirac matrices γa in the two-dimensional situation is determined by the Pauli matrices γa = (σ3, i σ1, i s σ2),[50] where the parameter s denotes the symbol ± (+ denotes spin up and – denotes spin down). According to the line element, the tetrads can be written as

Solving the Maurer–Cartan structure equation, we obtain γμ Γμ = –γ1/2 r. The stationary state of the Kemmer equation is a four-constituent wave function, and it can be written as[44]

where ψD is the solution to the Dirac equation. In addition, the two-dimensional Kemmer equation with a magnetic field can be defined through the magnetic vector potential in the cosmic string spacetime. Therefore, the Kemmer equation considering the magnetic vector potential in the cosmic string background can be written as[44]

To solve the generalized Kemmer oscillator, we replace ∂1 with . The operator is chosen as . Hence, the Kemmer equation under the Dirac oscillator interaction can be written as

When we substitute Eq. (2) into Eq. (3), we gain the generalized Kemmer oscillator coupled equation

It is obvious that the component of wave function satisfy the relation ψ2 = ψ3. In this case, combining with the relevant equation we can obtain

Meanwhile, we assume the wave function ψ2 = e–i λ φ ζ(r). Therefore, we have

In this present work, our main purpose is to solve Eq. (14) for some appropriate potential f(r).

3. The exact solution to the radial equation
3.1. The Kemmer equation’s solutions under the Morse potential

The Morse potential is a crucial molecular potential, which can describe the interaction between two atoms. In this section, we discuss that the potential function f(r) is Morse potential,[51,52]

where Δ0 and β are real constants describing property of the generalized Morse potential. Then the equation describing the generalized Kemmer oscillator in the cosmic string background with the magnetic field can be written as

We assume the wave function in the form of

Thus, equation (16) becomes

There is no doubt that the equation cannot be exactly solved due to existing the centrifugal term. To obtain the analytical solution, we can use some approximation approaches for the centrifugal term potential. We make the centrifugal term read

We also introduce a new variable ρ = e–β r, then equation (18) becomes

where

Equation (20) is a very user-friendly form, which can be solved through the so-called parametric NU method. Comparing it with the NU method, we can obtain the parameters

Using the NU method, the energy spectrum can be written as

At this moment, inserting detailed Σ1, Σ2, Σ3, the energy spectrum can be rewritten as

It should be noted that the magnetic field strength may eliminate the impact from the cosmic string spacetime and the magnetic field satisfies the condition . Therefore, the energy spectrum and wave function can read

where N is the normalization constant, and is Jacobi polynomials.

3.2. The Kemmer equation’s solutions under the Coulomb-liked potential

Let us consider the f(r) being the Coulomb-liked potential,[53–55]

where Δ1, Δ2, and Δ3 are real parameters. Inserting the Coulomb-liked potential into the equation, we can obtain

where

Further we choose the following ansatz:

Using the redefinition , equation (31) can be transformed to

The equation has the same form as the biconfluent Heun equation.[56–61] Therefore, ζ (ρ), through the Frobenius method, can be written as

We can also obtain the following recurrence relations:

Furthermore, according to the Ref. [62], the biconfluent Heun function can be expressed as

where and the coefficients Aj obey

In this case, the wave function can be written as

We can see that the biconfluent Heun functions at infinity are divergent. As a result, in order to enable the function having the n-degree multinomial form, it must meet the following relations:

where the multinomial An + 1 has n + 1 real roots when , and . Meanwhile, the multinomial An + 1 is the (n + 1)-dimensional determinant, which can be written as

where

In addition, we can obtain the energy spectrum

From the above spectrum, we can see that changing the intensity ΦB of the external magnetic field can eliminate the cosmic string spacetime effect, and the corresponding energy spectrum can be written as

Further, when the potential parameter Δ1 = 0, the system reduces to Cornell potential which is Coulomb potential plus linear potential, and the corresponding energy spectrum is similar to Ref. [43].

4. Conclusion

We have introduced a (1+2)-dimensional generalized Kemmer oscillator under uniform magnetic cosmic string background. The (1+2)-dimensional generalized Kemmer oscillator under uniform magnetic cosmic string background has been solved. The wave functions and energy eigenvalues have been presented using the NU method and the biconfluent Heun functions for the Morse potential and the Coulomb-liked potential, respectively. It is demonstrated that the wave function and energy spectrum have the dependence on angular deficit α, which is associated with the linear mass density of the cosmic string.

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