the total magnetization is proportional to the Bloch vector . All that is left to obtain the final form of the optical Bloch equations is the inclusion of the phenomenological relaxation terms. As a final aside, the above equation can be derived by considering the time evolution of the angular momentum operator in the Heisenberg picture. When coupled with the fact that , this equation is the same equation as before. Validity Two-state systems are the simplest non-trivial quantum systems that occur in nature, but the above-mentioned methods of analysis are not just valid for simple two-state systems. Any general multi-state quantum system can be treated as a two-state system as long as the observable one is interested in has two eigenvalues. For example, a spin-1/2 particle may in reality have additional translational or even rotational degrees of freedom, but those degrees of freedom are irrelevant to the preceding analysis. Mathematically, the neglected degrees of freedom correspond to the degeneracy of the spin eigenvalues. Another case where the effective two-state formalism is valid is when the system under consideration has two levels that are effectively decoupled from the system. This is the case in the analysis of the spontaneous or stimulated emission of light by atoms and that of charge qubits. In this case it should be kept in mind that the perturbations (interactions with an external field) are in the right range and do not cause transitions to states other than the ones of interest. Significance and other examples Pedagogically, the two-state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems. It can be used to illustrate fundamental quantum mechanical phenomena such as the interference exhibited by particles of the polarization states of the photon, but also more complex phenomena such as neutrino oscillation or the neutral K-meson oscillation. Two-state formalism can be used to describe simple mixing of states, which leads to phenomena such as resonance stabilization and other level crossing related symmetries. Such phenomena have a wide variety of application in chemistry. Phenomena with tremendous industrial applications such as the maser and laser can be explained using the two-state formalism. The two-state formalism also forms the basis of quantum computing. Qubits, which are the building blocks of a quantum computer, are nothing but two-state systems. Any quantum computational operation is a unitary operation that rotates the state vector on the Bloch