# Phase factor

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For any complex number written in polar form (such as reiθ), the phase factor is the exponential part (eiθ). As such, the term "phase factor" is similar to the term phasor, although the former term is more common in quantum mechanics. This phase factor is itself a complex number of absolute value 1. The variable θ appearing in such an expression is generally referred to as the phase. Multiplying the equation of a plane wave Aei(k·rωt) by a phase factor shifts the phase of the wave by θ:

$\text{e}^{i\theta}A\text{e}^{i\left({\mathbf{k}\cdot\mathbf{r}-\omega t}\right)} = A\text{e}^{i\left({\mathbf{k}\cdot\mathbf{r}-\omega t + \theta}\right)}$.

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 $\langle\phi|e^{-i\theta} A e^{i\theta}|\phi\rangle$ are the same.[1] 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.

## Notes

1. ^ Messiah (1999, p. 296)