Instantaneous angular velocity of quantum evolution

We introduce a metric in a geometric way for measuring the distance between two quantum states and derive the instantaneous angular velocity of a quantum state's dynamical evolution, which has two components: ΔEₜ/ in the longitudinal direction and Ēₜ/ in the latitudinal direction. By applying the instantaneous angular velocities in the dynamical evolution governed by a time-independent Hamiltonian, the fastest way for a pure state to evolve to a target state is discussed. As a consequence, we generalize the Mandelstam-Tamm and Margolus-Levitin bounds without using the Heisenberg uncertainty relation and point out the conditions for saturating the two bounds. The fastest dynamical evolution of a quantum system with a particular time-dependent Hamiltonian is also briefly discussed. The quantum features in the dynamical evolution are revealed through phase change and lifetime of the initial state.