Phase factor

For any complex number written in polar form (such as $r e^{iθ}$), the phase factor is the complex exponential ($e^{iθ}$), where the variable $θ$ 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 $Ae^{i(k·r − ωt)}$ by a phase factor $r e^{iθ}$ shifts the phase of the wave by $θ$: $$e^{i\theta} A\,e^{i({\mathbf{k}\cdot\mathbf{r} - \omega t})} = A\,e^{i({\mathbf{k}\cdot\mathbf{r}-\omega t + \theta})}.$$

In quantum mechanics, a phase factor is a complex coefficient $e^{iθ}$ that multiplies a ket $$|\psi\rangle$$ or bra $$\langle\phi|$$. It does not, in itself, have any physical meaning, since the introduction of a phase factor does not change the expectation values of a Hermitian operator. That is, the values of $$\langle\phi| A |\phi\rangle$$ and $$\langle\phi| A e^{i\theta} |\phi\rangle$$ are the same. However, differences in phase factors between two interacting quantum states can sometimes be measurable (such as in the Berry phase) and this can have important consequences. In optics, the phase factor is an important quantity in the treatment of interference.