Factorization of the evolution operator for a spin-orbit coupled system in an external magnetic field for an arbitrary L and S = ½

Abstract

The evolution operator of the spin-orbit coupled system in an external magnetic field is factorized according to the operator equation: exp it/h ( H_SO + H_Z) = exp( it/h H_SO) exp (it/h H_Z) C(t) where H_SO is the spin-orbit coupled Hamiltonian and H_Z is the Zeeman Hamiltonian. The preceding equation is expressed in matrix form using the theory of the O(4) group representations. For the special case of S = ½, an explicit form of the matrix C(t) is found. The operator corresponding to the matrix C(t) is found in a closed form using the projection operators constructed in the Appendix. Two special cases, corresponding to a weak and a strong external field, are considered, and possible applications of the results obtained are indicated.