Throughout this section, the reduced Planck constant  for convenience.
The product operator formalism is usually applied to sets of spin-1/2 particles, since the fact that the individual operators satisfy , where  is the identity operator, makes the commutation relations of product operators particularly simple. In principle the formalism could be extended to higher spins, but in practice the general irreducible spherical tensor treatment is more often used. As such, we consider only the spin-1/2 case below.
The main idea of the formalism is to make it easier to follow the system density operator , which evolves under a Hamiltonian  according to the Liouville-von Neumann equation as
For a time-independent Hamiltonian, the density operator inherits its solutions from the Schrödinger time-evolution operator  as