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For any complex number written in polar form (such as *r e ^{iθ}*), the

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:

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.

In optics, the phase factor is an important quantity in the treatment of interference.

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

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