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| {{Refimprove|date=December 2009}}
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| For any [[complex number]] written in polar form (such as ''r''[[e (mathematical constant)|e]]<sup>[[imaginary unit|i]]''θ''</sup>), the '''phase factor''' is the exponential part (e<sup>i''θ''</sup>). 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 (waves)|phase]]. Multiplying the equation of a [[plane wave]] ''A''e<sup>i('''k·r'''−''ωt'')</sup> by a phase factor [[phase shift|shifts the phase]] of the wave by ''θ'':
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| :<math>\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)}</math>.
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| In [[quantum mechanics]], a phase factor is a complex coefficient e<sup>i''θ''</sup> that multiplies a [[Bra-ket notation|ket]] <math>|\psi\rangle</math> or [[Bra-ket notation|bra]] <math>\langle\phi|</math>. 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 <math>\langle\phi|A|\phi\rangle</math> and <math>\langle\phi|e^{-i\theta} A e^{i\theta}|\phi\rangle</math> are the same.<ref>{{harvtxt|Messiah|1999|p=296}}</ref> However, ''differences'' in phase factors between two interacting [[quantum state]]s can sometimes be measurable (such as in the [[Berry phase]]) and this can have important consequences.
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| In [[optics]], the phase factor is an important quantity in the treatment of [[Interference (wave propagation)|interference]].
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| ==See also==
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| * [[Berry phase]]
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| * [[Bra-ket notation]]
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| * [[Euler's formula]]
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| * [[Phasor]]
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| * [[Plane wave]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|last=Messiah|first=Albert|authorlink=Albert Messiah|title=Quantum Mechanics|publisher=Dover|year=1999|isbn=0-486-40924-4}}
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| {{DEFAULTSORT:Phase Factor}}
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| [[Category:Quantum mechanics]]
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| [[Category:Information theory]]
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| [[Category:Quantum information science]]
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| [[Category:Notation]]
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