Fermi field and Dirac oscillator in a Som–Raychaudhuri space-time

Abstract

We investigate the relativistic dynamics of a Dirac field in the Som–Raychaudhuri space-time, which is described by a Gödel-type metric and a stationary cylindrical symmetric solution of Einstein field equations for a charged dust distribution in rigid rotation. In order to analyze the effect of various physical parameters of this space-time, we solve the Dirac equation in the Som–Raychaudhuri space-time and obtain the energy levels and eigenfunctions of the Dirac operator by using the Nikiforov–Uvarov method. We also examine the behaviour of the Dirac oscillator in the Som–Raychaudhuri space-time, in particular, the effect of its frequency and the vorticity parameter.

Brief review of the Nikiforov–Uvarov (NU) method

The Nikiforov–Uvarov method is helpful in order to find eigenvalues and eigenfunctions of the Schrödinger equation, as well as other second-order differential equations of physical interest. More details can be found in Refs. [75, 76]. According to this method, the eigenfunctions and eigenvalues of a second-order differential equation with potential

$$\begin{aligned} \frac{d^2\psi (s)}{ds^2}+\frac{\alpha _1-\alpha _2s}{s(1-\alpha _3s)}\frac{d\psi (s)}{ds}+\left[ \frac{-\beta _1s^2+\beta _2s-\beta _3}{s^2(1-\alpha _3s)^2}\right] \psi (s)=0 \end{aligned}$$

(A.1)

are given by

$$\begin{aligned} \psi \left( s \right) = {s^{{\alpha _{12}}}}{\left( {1 - {\alpha _3}s} \right) ^{ - {\alpha _{12}} - \frac{{{\alpha _{13}}}}{{{\alpha _3}}}}}{P_n}^{\left( {{\alpha _{10}} - 1,\frac{{{\alpha _{11}}}}{{{\alpha _3}}} - {\alpha _{10}} - 1} \right) }\left( {1 - 2{\alpha _3}s} \right) \end{aligned}$$

(A.2)

and

$$\begin{aligned}&n\alpha _2-(2n+1)\alpha _5+(2n+1)\left( \sqrt{\alpha _9}+\alpha _3\sqrt{\alpha _8}\right) +n(n-1)\alpha _3+\alpha _7\nonumber \\&\quad +\,2\alpha _3\alpha _8+2\sqrt{\alpha _8\alpha _9}=0, \end{aligned}$$

(A.3)

respectively. The parameters $\alpha _4$,..., $\alpha _{13}$ are obtained from the six parameters $\alpha _1$,..., $\alpha _3$, $\beta _1$,...,$\beta _3$ from Eq. (A.1) as follows:

$$\begin{aligned} \begin{array}{l} \alpha _4=\frac{1}{2}(1-\alpha _1),\quad \alpha _5=\frac{1}{2}(\alpha _2-2\alpha _3),\quad \alpha _6=\alpha _5^2+\beta _1,\quad \alpha _7=2\alpha _4\alpha _5-\beta _2,\\ \alpha _8=\alpha _4^2+\beta _3,\quad \alpha _9=\alpha _3\alpha _7+\alpha _3^2\alpha _8+\alpha _6,\quad \alpha _{10}=\alpha _1+2\alpha _4+2\sqrt{\alpha _8},\\ \alpha _{11}=\alpha _2-2\alpha _5+2\left( \sqrt{\alpha _9}+\alpha _3\sqrt{\alpha _8}\right) ,\quad \alpha _{12}=\alpha _4+\sqrt{\alpha _8},\quad \\ \alpha _{13}=\alpha _5-\sqrt{\alpha _9}-\alpha _3\sqrt{\alpha _8}. \end{array} \end{aligned}$$

(A.4)

In the particular case where $\alpha _3=0$, as in our case, we find

$$\begin{aligned} \mathop {\lim }\limits _{{\alpha _3} \rightarrow 0} {P_n}^{\left( {{\alpha _{10}} - 1,\frac{{{\alpha _{11}}}}{{{\alpha _3}}} - {\alpha _{10}} - 1} \right) }\left( {1 - 2{\alpha _3}s} \right) = L_n^{{\alpha _{10}} - 1}({\alpha _{11}}s), \end{aligned}$$

(A.5)

and the wave-function becomes

$$\begin{aligned} \psi (s)=s^{\alpha _{12}}\exp \left( \alpha _{13}s\right) \; L_n^{\alpha _{10}-1}\left( \alpha _{11}s\right) , \end{aligned}$$

(A.6)

where $L_n^{(\alpha )}$ denotes the generalized Laguerre polynomial.