  The time-dependent Schrodinger equation of a many particle spin system
consisting of an electron in a quantum dot interacting with the spins of the
nuclei (N) in the dot due to hyperfine interaction is solved exactly for a
given arbitrary initial state. The electron spin dynamics is then expressed in
terms of the reduced density matrix of the composite system by computing the
marginal density matrix of the electron. This is accomplished by classifying
states of the system by the total spin of the coupled electron and nuclear
system that commutes with the system Hamiltonian. These states are used in
enumerating and finding the exact solution of the time-dependent Schrodinger
equation. In each sector of the total spin, the problem reduces to solving a
linear simultaneous set of equations that are solved by matrix inversion. Such
solutions enable one to make a reliable approximate scheme for purposes of
numerical estimation of the various physical quantities. A methematical and
physical discussion of this procedure with the density matrix approach to this
problem is given here. To elucidate the procedure and its advantages, special
cases of N=3,4 are given in some detail in an Appendix.
