Phase operator for two-state systems

A Hermitean phase operator is constructed for two-state systems by means of an antiunitary transformation. The phase information is obtained from this operator by calculating a "phase vector," in close analogy to the way the vector of expectation values (Bloch vector) is calculated from the projection operator of a state. This construction is applied to two issues of quantum physics where phases are of crucial importance, i.e., the superposition principle and geometric phases. It is shown by these examples how the phase vector representation of these issues, offering a very useful visualization, derives from the algebraic properties of the phase operator.