This chapter describes the quantum spin system. Various model Hamiltonians are described, the history of previous solution attempts for special forms of these Hamiltonians (quantum and classical) is briefly outlined. The Heisenberg model (spin dimension of three) with a fully cross-coupled set of spins (hypersimplex spatial structure, or infinite space dimension), a uniform coupling constant, and zero external field is taken as being a good non-trivial candidate for exact solution. Failing to find an exact solution, a numerical solution is presented. The equilibrium solution is explored. A classical system which reproduces the equations of motion of the quantum system is explored. The the entropy, moment, correlation, and information correlation functions for the quantum Heisenberg system are discussed.
The quantum spin system is described by a Hamiltonian of the form
where the individual spin vectors in the system are labeled with a subscript.
In general the spin-spin coupling 
is a tensor and the external field 
is a vector, indicating that the coupling between the spins and between the field and the spins in different spatial directions may not be constant. Further, the subscripts on and indicate that the coupling between the spins and between the field and the spins may vary depending on their spatial organization. In the next two sections we discuss the various special forms that this Hamiltonian may take, the existing known solutions of these forms, and the dynamical equations that accompany this Hamiltonian.