  Transformation of Coulomb interaction terms to the pseudo-orbital basis
constituted by $J=1/2$ and $3/2$ states arising from spin-orbit coupling
provides a versatile tool. This formalism is applied to investigate magnetic
anisotropy effects on low-energy spin-wave excitations as well as high-energy
spin-orbit exciton modes in $\rm Sr_2 Ir O_4$. The Hund's coupling term
explictly yields easy-plane anisotropy, resulting in gapless (in-plane) and
gapped (out-of-plane) modes, in agreement with recent resonant inelastic x-ray
scattering (RIXS) measurements. The collective mode of inter-orbital,
spin-flip, particle-hole excitations with appropriate interaction strengths and
renormalized spin-orbit gap yields two well-defined propagating spin-orbit
exciton modes, with energy scale and dispersion in good agreement with RIXS
studies.
