Tensor Operator Algebra in Free Atoms and Ions

Abstract

When considering atomic problems it is usually convenient to operate in spherical polar coordinates rather than in Cartesian coordinates. The spherical harmonics Y _jm = |jm〉 then arise in a natural way as the simultaneous eigenfunctions of the Hamiltonian and the operators J ² and J _z of the square of the total angular momentum and of its z component. This fact is intimately related to the transformation properties of the |jm〉 eigenvectors under rotations. The rotation through the infinitesimal angle ε about the z axis of the physical system may be described by the operator^6,8

$${R_z}(\varepsilon ) = 1 - i\varepsilon {J_z}$$

(2.1)

.