Phase factor

For any complex number written in polar form (such as re^iθ), the phase factor is the complex exponential factor (e^iθ). 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.

The variable θ appearing in such an expression is generally referred to as the phase. Multiplying the equation of a plane wave Ae^{i(k·r−ωt)} 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 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 $\langle\phi|e^{-i\theta} A e^{i\theta}|\phi\rangle$ 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.

See also

• Berry phase
• Bra-ket notation
• Euler's formula
• Phasor
• Plane wave

References

• Messiah (1999, p. 296)