Factorization Method in Oscillator with the Aharonov-Casher System

We review the oscillator with Aharonov-Casher system and study some mathematical foundation about factorization method.The factorization method helps us to obtain the energy spectrum and general wave function for the corresponding system in some spin condition. The factorization method leads us to obtain the raising and lowering operators for the Aharonov-Casher system. The corresponding operators give us the generators of the algebra.


Introduction
As we know, the relativistic quantum dynamics of a natural particle which describes the relativistic Aharonov-Casher system [1] is given by introducing minimal coupling into the Dirac equation [2,3] in Cartesian coordinates which is given by ∑ ] () , where we consider the natural units ℎ =  = 1.Also,  corresponds to the magnetic dipole moment of natural particle, and  ] () corresponds to the electromagnetic tensor, which are defined by  0 = − 0 =   and   = −  = −    and ∑  = (/2)[  ,   ].The   matrices correspond to the Dirac matrices in Minkowski space-time [4]: with     +     = −2  , with ⃗ ∑ being the spin vector and   being Pauli matrices.The tensor   = diag(− + ++) is the Minkowski tenor.Moreover, by introducing the coupling that describes the Dirac oscillator ⃗  → ⃗  −  β ρ into the nonminimal coupling (1), we can see that the whole system is cylindrically symmetric.So, we can work with curvilinear coordinates  =  cos  and  =  sin .Therefore, we write the line element of the Minkowski space-time in the following form: (3) Here, we note that, in curvilinear coordinate (both flat and curved space-time background), the relativistic quantum dynamics of a neutral particle with a permanent magnetic dipole moment interacting with the external field is not described by the Dirac equation with the introduction of nonminimal coupling (1) anymore.Based on the spinor theory in curved space-time, nonminimal coupling (1) plus the coupling describing the Dirac oscillator becomes where ∇  +   + Γ  () corresponds to the components of the covariant derivative of a spinor, with Γ  = (/4)  () ∑  being the spinnorial connection [5,6], and In the spinor theory in curved space-time, the   matrices are defined in the local reference frame of the observers and are identical to the Dirac matrices defined in Minkowski space-time (3).In this notation, the indices (, ,  = 0, 1, 2, 3) indicate the local reference frame, while the indices (, ]) indicate the space-time indices.Thus, the   matrices given by ( 4) are related to the   via   =    ()  , where components    () are called tetrades and give rise to the local reference frame of the observers.The tetrades satisfy the following equation [5,6]: The tetrades also have an inverse defined as   =    ()  , where Furthermore, the components of the spinnorial connection can be obtained by solving the Cartan structure [6] in the absence of torsion: where Here, we consider radial electric field as ⃗  = (/) ρ and also consider the magnetic dipole moment parallel to the -axis.
In that case we can rewrite the Dirac equation as where [1].In order to solve (10) one can write Ψ in terms of two components of spinors as where  and  are spinors of two components.We substitute (11) in (10) to obtain two coupled equations for  and .Now, we are going to write the first coupled equation which is given by and the second coupled equation is By using ( 12) and ( 13) one can obtain the following second order differential equation: Here,  is eigenfunction of the Pauli matrix  3 and total angular momentum Ĵ = −  , and the -component of the momentum p = −  commutes with the Hamiltonian of (14).In that case, we can write the solution of (14) in terms of the eigenvalues of the operators p = −  and Ĵ = −  , = ±1 give  3  0 =  0 , where  0 = ( + ,  − )  , with ℓ = 0, ±1, ±2, . . .,  being constant and  being normalization factor.Thus, substituting the solution (15) into the second order differential equation ( 14), we obtain the following radial equation: where Advances in Mathematical Physics 3 In order to solve (16) we change variables given by  =  2 and obtain Again, we choose the change of variable as follows: so, (18) changes as follows: In order to obtain   () in ( 19), we have to compare (20) with known polynomial.For this reason, first we introduce the Laguerre polynomial which is corresponding to (20).So, in this paper first we will try to review some mathematical foundation about factorization method [7].In Section 3 we take advantage of factorization method and obtain the energy spectrum and general wave function for the corresponding system.Also, we show that the corresponding equation can be factorized first with respect to   and then with respect to  and   .These lead us to obtain the raising and lowering operators.Note that the shape invariant equation ( 27) can be written as the raising and lowering relations for the Aharonov-Casher system.These operators will be generators algebra.

Mathematical Foundation
Using the factorization approach, we compute the energy spectrum  and also bound states Ψ through the comparison of the differential equation given in (18) with associated Laguerre differential equation in an appropriate manner.We also factorize the second order differential equations into new sets of operators  − ,  + and shape invariant form, which are the first order differential equations.This process is called factorization method.Before anything else we will try to explain the associated Laguerre differential equation  (,) ,  () in factorization method point of view.To start, we need to recall that, for the real parameters  > −1 and  > 0, the associated Laguerre differential equation corresponding to  (,) ,  () in the interval  ∈ (0, ∞) is introduced as [ where operators  +   () and  −   () are given by the following equation: One may write down the shape invariance equation ( 23) as the raising and lowering relations: where the differential operators are functions of parameters  and   which are obtained as Note that the shape invariant equation ( 28) can be written as the raising and lowering relations

Advances in Mathematical Physics
The above mentioned method with some calculations leads to the following normalization coefficient: Also, the normalization coefficient equation (29) has been so chosen that the associated Laguerre functions  (,) ,  (), with the same   but with different  with respect to the inner product with the weight functions    − , form an orthonormal set in the interval 0 ≤  < ∞: Now, we go back to the Dirac oscillator with Aharonov-Casher system and compare it with associated Laguerre equation.
Here, we note that the factorization method and shape invariance condition help us to factorize the second order oscillator with the Aharonov-Casher equation.It means that such equation factorized in terms of two first order operators which are known by raising and lowering operators.