Dmitry Uskov and Ravi Rau have developed a very compact solution of time dependent operator equations in quantum physics. A general problem for N quantum states has been formulated as analogous to one with two states. The N dimensions are viewed as split into (N-1) and one dimensions, and effective Hamiltonians derived for each of those two sets. Thereby, a stepwise solution is provided for any N. A specific focus is on phases of the evolution operator and their decomposition into dynamical and geometrical contributions, these considerations of interest today in the fields of quantum computation and quantum information. Among recent papers submitted for publication is one which draws parallels between effective Hamiltonians for time-dependent and time-independent problems in quantum physics.