This article needs additional citations for verification. (December 2009) (Learn how and when to remove this template message) |

For any complex number written in polar form (such as *r*e^{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 *A*e^{i(k·r−ωt)} by a phase factor shifts the phase of the wave by *θ*:

- .

In quantum mechanics, a phase factor is a complex coefficient e^{iθ} that multiplies a ket or bra . 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 and 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.

## See also

## Notes

**^**Messiah (1999, p. 296)

## References

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