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For anycomplex number written inpolar form (such asre^iθ), thephase factor is thecomplex exponential (e^iθ), where the variableθ is thephase of a wave or other periodic function. The phase factor is aunit complex number, i.e. a complex number ofabsolute value1. It is commonly used inquantum mechanics andoptics. It is a special case ofphasors, which may have arbitrary magnitude (i.e. not necessarily on the unit circle in thecomplex plane).

Multiplying the equation of aplane waveAe^{i(k·r −ωt)} by a phase factorr e^iθshifts the phase of the wave byθ:$${\displaystyle e^{i\theta }A{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}=A{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t+\theta )}.}$$

In quantum mechanics, a phase factor is a complex coefficiente^iθ that multiplies aket${\displaystyle |\psi ⟩}$ orbra${\displaystyle ⟨\varphi |}$. It does not, in itself, have any physical meaning, since the introduction of a phase factor does not change the expectation values of aHermitian operator. That is, the values of${\displaystyle ⟨\varphi |A|\varphi ⟩}$ and${\displaystyle ⟨\psi |A|\psi ⟩}$, where${\displaystyle |\psi ⟩=e^{i\theta }|\varphi ⟩}$, are the same.^[1] However,differences in phase factors between two interactingquantum states can sometimes be measurable (such as in theBerry phase) and this can have important consequences.In optics, the phase factor is an important quantity in the treatment ofinterference.

• Berry phase
• Bra-ket notation
• Euler's formula
• Phasor
• Plane wave
• Thecircle group U(1)

• Messiah, Albert (1999),Quantum Mechanics, Dover,ISBN 0-486-40924-4

• Information theory
• Quantum information science
• Notation

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