  An analytical formula is obtained to describe the evolution of the average
populations of spin components of spin-1 atomic gases. The formula is derived
from the exact time-dependent solution of the Hamiltonian $H_{S}=c
mathbf{S}^{2}$ without using approximation. Therefore it goes beyond the mean
field theory and provides a general, accurate, and complete description for the
whole process of non-dissipative evolution starting from various initial
states. The numerical results directly given by the formula coincide
qualitatively well with existing experimental data, and also with other
theoretical results from solving dynamic differential equations. For some
special cases of initial state, instead of undergoing strong oscillation as
found previously, the evolution is found to go on very steadily in a very long
duration.
