polarization properties of the photon (it is always orthogonal to $ {\bf k} $; the radiation is treated in the temporal gauge); the vector quantity $ {\langle}{\boldsymbol}\alpha {\rangle}$ is related to the transition amplitude: $$\label{rad_alpha_def}
    {\langle}{\boldsymbol}\alpha {\rangle}=
        \int d^{3}x \, \Psi'^{\dag} \left( {\boldsymbol}\alpha e^{-i {\bf kx}} \right) \Psi.$$

Let $ (\theta, \varphi) $ be the angles characterizing the direction of the radiation of a given polarization in a spherical coordinate system with the $ z $-axis parallel to the magnetic field orientation, so that $$\label{rad_k}
    {\bf k} =
        k \{ \sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta \}.$$ Evaluating the integral in (\[rad\_alpha\_def\]), due to the general form of the wave functions (\[sol\_Psi\_phi\]), (\[sol\_phi\]), one finds (see [@book_hep_rad] for the details of these calculations): $${\langle}{\boldsymbol}\alpha {\rangle}\ =
        {\langle}\bar{{\boldsymbol}\alpha} {\rangle}\, I_{s,s'}(x) \, \delta( p' - p + k \cos \theta ),$$ where $$\label{rad_alpha}
    \begin{aligned}
        -i {\langle}\bar{\alpha}_{1} {\rangle}\ & = &&
            ( c_{1}'^{*} c_{4}^{} + c_{3}'^{*} c_{2}^{} ) I_{n,n'-1}(x)
            && - && ( c_{1}^{} c_{4}'^{*} + c_{3}^{} c_{2}'^{*} ) I_{n-1,n'}(x), \\
        {\langle}\bar{\alpha}_{2} {\rangle}\ & = &&
            ( c_{1}'^{*} c_{4}^{} + c_{3}'^{*} c_{2}^{} ) I_{n,n'-1}(x)
            && + && ( c_{1}^{} c_{4}'^{*} + c_{3}^{} c_{2}'^{*} ) I_{n-1,n'}(x), \\
        {\langle}\bar{\alpha}_{3} {\rangle}\ & 