# Phase factor

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} For any complex number written in polar form (such as reiθ), the phase factor is the complex exponential factor (eiθ). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e., not necessarily part of the circle group). The phase factor is a unit complex number, i.e., of absolute value 1. It is commonly used in quantum mechanics.
${\text{e}}^{i\theta }A{\text{e}}^{i\left({\mathbf {k} \cdot \mathbf {r} -\omega t}\right)}=A{\text{e}}^{i\left({\mathbf {k} \cdot \mathbf {r} -\omega t+\theta }\right)}$ .
In quantum mechanics, a phase factor is a complex coefficient eiθ 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 $\langle \phi |e^{-i\theta }Ae^{i\theta }|\phi \rangle$ 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.