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| {{Technical|date=March 2012}}
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| In [[quantum mechanics]], [[momentum]] (like all other physical variables) is defined as an [[operator (physics)|operator]], which "acts on" or pre-multiplies the [[wave function]] {{math|ψ('''r''', ''t'' )}} to extract the momentum [[eigenvalue]] from the wavefunction: the momentum vector a particle would have when [[Measurement in quantum mechanics|measured in an experiment]]. The '''momentum operator''' is an example of a [[differential operator]].
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| At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including [[Niels Bohr]], [[Arnold Sommerfeld]], [[Erwin Schrödinger]], and [[Eugene Wigner]].
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| ==Origin from De Broglie plane waves==
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| The momentum and energy operators can be constructed in the following way.<ref>Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0</ref>
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| ;One dimension
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| Starting in one dimension, using the [[plane wave]] solution to [[Schrödinger's equation]]:
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| :<math> \Psi = e^{i(kx-\omega t)} \,\!</math>
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| The first order [[partial derivative]] with respect to space is
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| :<math> \frac{\partial \Psi}{\partial x} = i k e^{i(kx-\omega t)} = i k \Psi \,\!</math>
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| By expressing of {{mvar|k}} from the [[De Broglie relation]]:
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| :<math> p=\hbar k\,\!</math>
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| the formula for the derivative of {{math|ψ}} becomes:
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| :<math> \frac{\partial \Psi}{\partial x} = i \frac{p}{\hbar} \Psi \,\!</math>
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| This suggests the operator equivalence:
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| :<math> \hat{p} = -i\hbar \frac{\partial }{\partial x} \,\!</math>
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| so the momentum value {{mvar|p}} is a [[Scalar (mathematics)|scalar]] factor, the momentum of the particle and the value that is measured, is the [[eigenvalue]] of the operator.
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| Since the partial derivative is a [[linear operator]], the momentum operator is also linear, and because any wavefunction can be expressed as a [[quantum superposition|superposition of other states]], when this momentum operator acts on the entire superimposed wave, it yields the momentum eigenvalues for each plane wave component, the momenta add to the total momentum of the superimposed wave.
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| ;Three dimensions
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| The derivation in three dimensions is the same, except the gradient operator [[del]] is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is:
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| :<math> \Psi = e^{i(\bold{k}\cdot\bold{r}-\omega t)} \,\!</math>
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| and the gradient is
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| :<math> \begin{align} \nabla \Psi & = \bold{e}_x\frac{\partial \Psi}{\partial x} + \bold{e}_y\frac{\partial \Psi}{\partial y} + \bold{e}_z\frac{\partial \Psi}{\partial z} \\
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| & = i k_x\Psi\bold{e}_x + i k_y\Psi\bold{e}_y+ i k_z\Psi\bold{e}_z \\
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| & = \frac{i}{\hbar} \left ( p_x\bold{e}_x + p_y\bold{e}_y+ p_z\bold{e}_z \right)\Psi \\
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| & = \frac{i}{\hbar} \bold{\hat{p}}\Psi \\
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| \end{align} \,\!</math>
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| where <math>\bold{e}_x</math>, <math>\bold{e}_y</math> and <math>\bold{e}_z</math> are the [[unit vector]]s for the three spatial dimensions, hence
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| :<math> \bold{\hat{p}} = -i \hbar \nabla \,\!</math>
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| This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables.
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| ==Definition (position space)==
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| {{see also|Position and momentum space}}
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| For a single particle with no [[electric charge]] and no [[spin (physics)|spin]], the momentum operator can be written in the position basis as:<ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref>
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| :<math>\bold{\hat{p}}=-i\hbar\nabla</math>
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| where ∇ is the [[gradient]] operator, ''ħ'' is the [[reduced Planck constant]], and {{math|''i''}} is the [[imaginary unit]].
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| In one spatial dimension this becomes:
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| :<math>\hat{p}=\hat{p}_x=-i\hbar{\partial \over \partial x}.</math>
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| This is a commonly encountered form of the momentum operator, though not the most general one. For a charged particle {{mvar|q}} in an [[electromagnetic field]], described by the [[scalar potential]] {{mvar|φ}} and [[vector potential]] {{math|'''A'''}}, the momentum operator must be replaced by:<ref>Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0</ref>
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| :<math>\bold{\hat{p}} = -i\hbar\nabla - q\bold{A} </math>
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| where the [[canonical momentum]] operator is the above momentum operator:
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| :<math>\bold{\hat{P}} = -i\hbar\nabla </math>
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| This is of course true for electrically neutral particles also, since the second term vanishes if {{mvar|q}} is zero and the original operator appears.
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| ==Properties==
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| ===Hermiticity===
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| The momentum operator is always a [[Hermitian operator]] when it acts on physical (in particular, [[Normalisable wave function|normalizable]]) quantum states.<ref>See [http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf Lecture notes 1 by Robert Littlejohn] for a specific mathematical discussion and proof for the case of a single, uncharged, spin-zero particle. See [http://bohr.physics.berkeley.edu/classes/221/1112/notes/spatialdof.pdf Lecture notes 4 by Robert Littlejohn] for the general case.</ref>
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| ===Canonical commutation relation===
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| {{further2|[[Canonical commutation relation]]}}
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| One can easily show that by appropriately using the momentum basis and the position basis:
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| :<math> \left [ \hat{ x }, \hat{ p } \right ] = \hat{x} \hat{p} - \hat{p} \hat{x} = i \hbar. </math>
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| The [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]] defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, [[position operator|position]] and momentum are [[canonical conjugate variables|conjugate variables]].
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| ===Fourier transform===
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| One can show that the [[Fourier transform]] of the momentum in [[quantum mechanics]] is the [[position (quantum mechanics)|position operator]]. The Fourier transform turns the momentum-basis into the position-basis. The following discussion uses the [[Bra-ket notation]]:
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| :<math> \langle x | \hat{p} | \psi \rangle = - i \hbar {d \over dx} \psi ( x ) </math>
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| The same applies for the [[Position operator]] in the momentum basis:
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| :<math> \langle p | \hat{x} | \psi \rangle = i \hbar {d \over dp} \psi ( p ) </math>
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| and other useful relations:
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| :<math> \langle p | \hat{x} | p' \rangle = i \hbar {d \over dp} \delta (p - p') </math>
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| :<math> \langle x | \hat{p} | x' \rangle = -i \hbar {d \over dx} \delta (x - x') </math>
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| where <math>\delta</math> stands for [[Dirac's delta function]].
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| ==Derivation from infinitesimal translations==
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| {{see also|Noether's theorem}}
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| The [[shift operator|translation operator]] is denoted {{math|''T''(''ϵ'')}}, where {{mvar|ϵ}} represents the length of the translation. It satisfies the following identity:
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| :<math> T(\epsilon) | \psi \rangle = \int dx T(\epsilon) | x \rangle \langle x | \psi \rangle </math>
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| that becomes
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| :<math> \int dx | x + \epsilon \rangle \langle x | \psi \rangle = \int dx | x \rangle \langle x - \epsilon | \psi \rangle = \int dx | x \rangle \psi(x - \epsilon) </math>
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| Assuming the function {{mvar|ψ}} to be [[analytic function|analytic]] (i.e. [[differentiable]] in some domain of the [[complex plane]]), one may expand in a [[Taylor series]] about {{mvar|x}}:
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| :<math>\psi(x-\epsilon) = \psi(x) - \epsilon {d \psi \over dx} </math>
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| so for [[infinitesimal]] values of {{mvar|ϵ}}:
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| :<math> T(\epsilon) = 1 - \epsilon {d \over dx} = 1 - {i \over \hbar} \epsilon \left ( - i \hbar{ d \over dx} \right )</math>
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| As it is known from [[classical mechanics]], the [[momentum]] is the generator of [[translation (physics)|translation]], so the relation between translation and momentum operators is:
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| :<math> T(\epsilon) = 1 - {i \over \hbar} \epsilon \hat{p}</math>
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| thus
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| :<math> \hat{p} = - i \hbar { d \over dx }\,. </math>
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| ==4-momentum operator==
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| Inserting the 3d momentum operator above and the [[energy operator]] into the [[4-momentum]] (as a [[1-form]] with (+−−−) [[metric signature]]):
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| :<math>P_\mu = \left(\frac{E}{c},-\bold{p}\right) \,\!</math>
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| obtains the '''4-momentum operator''';
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| :<math>\hat{P}_\mu = \left(\frac{1}{c}\hat{E},-\bold{\hat{p}}\right) = i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right) = i\hbar\partial_\mu \,\!</math>
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| where ∂<sub>μ</sub> is the [[4-gradient]], and the {{math|−''iħ'' }} becomes {{math|+''iħ'' }} preceding the 3-momentum operator. This operator occurs in relativistic [[quantum field theory]], such as the [[Dirac equation]] and other [[relativistic wave equations]], since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order [[partial derivative]]s for [[Lorentz covariance]].
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| The [[Dirac operator]] and [[Dirac slash]] of the 4-momentum is given by contracting with the [[gamma matrices]]:
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| :<math> \gamma^\mu\hat{P}_\mu = i\hbar \gamma^\mu\partial_\mu = \hat{P}\!\!\!\!/\ = i\hbar\partial\!\!\!/\ </math>
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| If the signature was (−+++), the operator would be
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| :<math>\hat{P}_\mu = \left(-\frac{1}{c}\hat{E},\bold{\hat{p}}\right) = -i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right) = -i\hbar\partial_\mu \,\!</math>
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| instead.
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| ==See also==
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| *[[Mathematical descriptions of the electromagnetic field]]
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| *[[Relativistic wave equations]]
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| *[[Pauli–Lubanski pseudovector]]
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| ==References==
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| {{reflist}}
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| {{Physics operator}}
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| [[Category:Quantum mechanics]]
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