Phase factor

In physics and representation theory, a phase factor is a multiplier representing the phase of a wave or the phase difference between two quantities. It is formulated as a unit complex number, that is a complex number with absolute value . For a complex number written in polar form, such as , the phase factor is the complex exponential, , where the variable is the phase and is the imaginary unit. If a quantity like a scalar, vector, or a matrix (representing a wave, state, or operator) is equal to another quantity times a phase factor, then those two quantities are said to be equivalent up to the phase factor, as it leaves the length (or norm) unchanged. As a set, the possible phase factors form the circle group U(1) , but the term often refers to a scalar recording a phase choice or convention, or an ambiguity in choosing a representative.

In physics and representation theory, a phase factor is a multiplier representing the phase of a wave or the phase difference between two quantities. It is formulated as a unit complex number, that is a complex number with absolute value . For a complex number written in polar form, such as , the phase factor is the complex exponential, , where the variable is the phase and is the imaginary unit. If a quantity like a scalar, vector, or a matrix (representing a wave, state, or operator) is equal to another quantity times a phase factor, then those two quantities are said to be equivalent up to the phase factor, as it leaves the length (or norm) unchanged. As a set, the possible phase factors form the circle group U(1) , but the term often refers to a scalar recording a phase choice or convention, or an ambiguity in choosing a representative. z = e^{i\theta} = \cos \theta + i \sin \theta. z^*z = 1. e^{i\theta} A\,e^{i({\mathbf{k}\cdot\mathbf{r} - \omega t})} = A\,e^{i({\mathbf{k}\cdot\mathbf{r} - \omega t + \theta})}. This phase factor is related to the arbitrary selection of the origin of the time axis. In quantum mechanics, a phase factor is a complex coefficient 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; this effect is known as phase ambiguity. That is, the values of \langle\phi| A |\phi\rangle and \langle\psi| A |\psi\rangle , where |\psi\rangle = e^{i\theta}|\phi\rangle , are the same. The phase ambiguity may also be described as a flexibility in the definition of quantum state functions. For example, the eigenfunctions of the angular momentum operator are uniquely defined "except for a phase factor". In defining spherical harmonics for use in quantum mechanics, the phase factor may be selected to have a standard value initially selected by Edward Condon and G.H. Shortley. For example, this convention is used for the Clebsch–Gordan coefficients. U(g)U(h) = \omega(g,h)U(gh), where \omega(g,h) is a phase factor. Such an action is a projective representation. A related ambiguity occurs in the representation theory of the Heisenberg group. Because of the Stone–von Neumann theorem, an automorphism of the underlying position-momentum space gives a unitary operator of the oscillator representation, but only up to a phase factor. The resulting operators therefore define a projective representation of the symplectic group. Passing to the metaplectic group resolves this ambiguity. ==See also== Berry phase Bra-ket notation Euler's formula Phasor Plane wave The circle group U(1) ==References==

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