† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11465006 and 11565009).
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.
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
The line element, in cylindrical coordinates, of cosmic string spacetime can be written as[45–47]
![](cpb_29_7_070302/cpb_29_7_070302_ieqn1.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn2.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn3.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn4.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn5.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn6.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn7.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn8.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn9.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn10.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn11.gif)
In this present work, our main purpose is to solve Eq. (
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]
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
![](cpb_29_7_070302/cpb_29_7_070302_ieqn12.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn13.gif)
Let us consider the f(r) being the Coulomb-liked potential,[53–55]
![](cpb_29_7_070302/cpb_29_7_070302_ieqn14.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn15.gif)
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:
![](cpb_29_7_070302/cpb_29_7_070302_ieqn16.gif)
![](cpb_29_7_070302/cpb_29_7_070302_ieqn17.gif)
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
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.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] | |
[42] | |
[43] | |
[44] | |
[45] | |
[46] | |
[47] | |
[48] | |
[49] | |
[50] | |
[51] | |
[52] | |
[53] | |
[54] | |
[55] | |
[56] | |
[57] | |
[58] | |
[59] | |
[60] | |
[61] | |
[62] |