For any complex number written in polar form (such as), the phase factor is the complex exponential, where the variable is the phase of a wave or other periodic function. The phase factor is a unit complex number, i.e. a complex number of absolute value 1. It is commonly used in quantum mechanics and optics. It is a special case of phasors, which may have arbitrary magnitude (i.e. not necessarily on the unit circle in the complex plane).
Multiplying the equation of a plane wave by a phase factor shifts the phase of the wave by :
|\psi\rangle
\langle\phi|
\langle\phi|A|\phi\rangle
\langle\phi|Aei\theta|\phi\rangle