ABSTRACT
This chapter represents an ensemble of identical spins using the density matrix formalism. It defines the matrix forms of the Cartesian operators that represent the Cartesian components of spin angular momentum, the total spin angular momentum, and operators representing connectivity between the stationary states of the system, so as to define the density matrix. In reality, very few spins in the ensemble are in "pure" states at any given time: there are constantly local perturbations in the electromagnetic environment that give rise to mixed or superposition states. The frequency distribution of such fluctuations is given by the spectral density function. For analyzing systems with more than one coupled spin, it is usually convenient to use the product operator formalism which is derived from the single-element basis set in order to calculate the effects of pulses and evolutions on one type of spin in the ensemble. The product operator formalism is extremely useful for describing multidimensional nuclear magnetic resonance (NMR) experiments.
