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Extra resources for Computation of atomic processes : a handbook for the ATOM programs
2. Representation of Multipoles and Statistical Tensors To obtain the statistical tensors of the photon, we consider the representation of the total angular momentum. 120) p=O,ILM Wave functions I pLM) describe the states of the photon with angular momentum L and its projection M on the quantization axis z. Summation in Eq. 120) starts from L = 1 (an absence of photons with L = 0 is a result of their transverse polarization). The quantum number p specifies the type of photon: either electric (p = 0) or magnetic (p = O.
110) The state of a photon linearly polarized along some direction cp in the xy-plane can be then written in the form Ik,eqJ) = cos cp Ik,ex) + sincp Ik,ey) 1 ( e-lqJlk,A=+I)-ezqJlk,A=-I) . 113) It follows from Eq. 115) Hence, in the case of arbitrary PI, P2, and P3, the photon is circularly and linearly polarized. 116) The equality ~}=l The parameter p? = 1 indicates that the polarization state of the photon is pure. 117) is called the degree of linear polarization. 116) imposes the restriction P; + S 1.
Under inversion, the statistical tensors acquire a phase factor 11 = nnf, where nand nf are internal parities of states with j and 1', respectively. If at least one of the states does not have definite parity, the transformation, which includes inversion, in general is not the transformation of symmetry. The reflection in a plane is a product of two transformations: rotation through an angle 1800 about the axis perpendicular to the plane and the inversion. Rotations and inversion are commutative.