A time-dependent quantum mechanical method for propagating the wave function on several electronic states is discussed for the polyatomic case and illustrated by the quenching collision of a Na (3p2P) atom by H2. The specification of method is governed by the need to have a clear physical interpretation of the results, by the recognition that the motion on a given electronic state can often (but not always) be well approximated by classical mechanics, and by the need for a computational procedure that is simple enough to handle polyatomic systems. These desiderata are realized by the spawning technique which is discussed in detail. One more feature of the method is that it allows for a smooth interface with the methodologies of quantum chemistry so that the electronic structure problem can be solved simultaneously with the time propagation of the nuclear dynamics. The method is derived from a variational principle and so can yield quantum mechanically numerically converged results. The parameters that govern the numerical accuracy of the method are explicitly discussed with special reference to their physical significance. The quenching of a Na (3p2P) atom by H2 due to a conical intersection of two potential energy surfaces is used as a computational example since it illustrates many of the features of the method. This collision is found to be sticky and exhibits many sequential nonadiabatic couplings, each of which is localized in time, where the quenching probability per traversal of the conical intersection region is small. However, the accumulated transfer of population to the ground state can be significant since the duration of the overall transfer is spread over many vibrational periods of H2.