Some analytical results forABC, ABCD, andXBCD coupled spin 1/2 systems. II

Abstract

In the preceding paper, it was shown that the calculation of the density matrixρ(t) for multiply connectedABC, etc., spin 1/2 spin systems can be greatly simplified by subdividing the Hamiltonian$H$ into$(H_1 + H_2 )$, where$H_1$ is a suitable linear combination of the constants of the motion. In this paper, a framework for the determination of the time evolution of high-order multipolar quantum states is presented and discussed. It is shown that the necessary mathematical labour is reduced to a minimum by (i) exploiting the fact that$J_z$ is a good quantum number, and (ii) using the theory of partitioned matrices. For example, it is shown that for a generaln-coupled spin 1/2 system, the spin dynamics of the$Q = \pm K_{max} ( \pm K_{max} \mp 1)$ multipolar states, whereK _max is the maximum tensorial rank, can be determined without the need to diagonalize the full 2ⁿ × 2ⁿ Hamiltonian matrix, wheren is the number of spins. In fact, to describe the time evolution of the$Q = ( \pm K_{max} \mp 1)$ multipolar states it is only necessary to diagonalize twon ×n matrices at most. Finally, some cautionary remarks are made concerning the use of the “weak-coupling approximation”.