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{{Short description|Operator in quantum mechanics}}
In [[quantum mechanics]], the '''momentum operator''' is the [[operator (physics)|operator]] associated with the [[momentum (physics)|linear momentum]]. The momentum operator is, in the position representation, an example of a [[differential operator]]. For the case of one particle in one spatial dimension, the definition is:
<math display="block">\hat{p} = - i \hbar \frac{\partial}{\partial x} </math>
where {{math|''ħ''}} is the [[reduced Planck constant]], {{math|''i''}} the [[imaginary unit]], {{mvar|x}} is the spatial coordinate, and a partial derivative (denoted by <math>\partial/\partial x</math>) is used instead of a [[total derivative]] ({{math|''d''/''dx''}}) since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows:
<math display="block"> \hat{p}\psi = - i \hbar \frac{\partial \psi}{\partial x} </math>

In a basis of [[Hilbert space]] consisting of momentum [[eigenstate]]s expressed in the momentum representation, the action of the operator is simply multiplication by {{mvar|p}}, i.e. it is a [[multiplication operator]], just as the [[position operator]] is a multiplication operator in the position representation. Note that the definition above is the [[canonical momentum]], which is not [[Gauge-invariant|gauge invariant]] and not a measurable physical quantity for charged particles in an [[electromagnetic field]]. In that case, the [[canonical momentum]] is not equal to the [[kinetic momentum]].

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]]. Its existence and form is sometimes taken as one of the foundational postulates of quantum mechanics.

== Origin from de Broglie plane waves ==
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>

=== One dimension ===
Starting in one dimension, using the [[plane wave]] solution to [[Schrödinger's equation]] of a single free particle,
<math display="block"> \psi(x, t) = e^{\frac{i}{\hbar}(px - Et)},</math>
where {{mvar|p}} is interpreted as momentum in the {{mvar|x}}-direction and {{mvar|E}} is the particle energy. The first order partial derivative with respect to space is
<math display="block"> \frac{\partial \psi(x, t)}{\partial x} = \frac{ip}{\hbar} e^{\frac{i}{\hbar}(px - Et)} = \frac{ip}{\hbar} \psi.</math>

This suggests the operator equivalence
<math display="block"> \hat{p} = -i\hbar \frac{\partial }{\partial x}</math>
so the momentum of the particle and the value that is measured when a particle is in a plane wave state is the [[Dirac_delta_function#Quantum_mechanics|(generalized) eigenvalue]] of the above operator.<ref>{{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree=  PhD |publisher= Universidad de Valladolid|page=106}}
</ref>

Since the partial derivative is a [[linear operator]], the momentum operator is also linear, and because any wave function 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. These new components then superimpose to form the new state, in general not a multiple of the old wave function.

=== Three dimensions ===
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:
<math display="block"> \psi = e^{\frac{i}{\hbar}(\mathbf{p}\cdot\mathbf{r}-E t)}</math>
and the gradient is
<math display="block"> \begin{align}
\nabla \psi &= \mathbf{e}_x\frac{\partial \psi}{\partial x} + \mathbf{e}_y\frac{\partial \psi}{\partial y} + \mathbf{e}_z\frac{\partial \psi}{\partial z} \\ 
& = \frac{i}{\hbar} \left ( p_x\mathbf{e}_x + p_y\mathbf{e}_y+ p_z\mathbf{e}_z \right)\psi \\
& = \frac{i}{\hbar} \mathbf{p}\psi
\end{align}</math>
where {{math|'''e'''<sub>''x''</sub>}}, {{math|'''e'''<sub>''y''</sub>}}, and {{math|'''e'''<sub>''z''</sub>}} are the [[unit vector]]s for the three spatial dimensions, hence
<math display="block"> \mathbf{\hat{p}} = -i \hbar \nabla</math>

This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables.

== Definition (position space) ==
{{see also|Position and momentum space}}

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|0-07-145546-9}}</ref>
<math display="block">\mathbf{\hat{p}}=-i\hbar\nabla</math>
where {{math|∇}} is the [[gradient]] operator, {{math|''ħ''}} is the [[reduced Planck constant]], and {{math|''i''}} is the [[imaginary unit]].

In one spatial dimension, this becomes<ref>In the position coordinate representation,  that is, <math display="inline">-i \hbar \int dx \left|x\right\rangle \partial_x \langle x|.</math></ref>
<math display="block">\hat{p}=\hat{p}_x=-i\hbar{\partial \over \partial x}.</math>

This is the expression for the [[Canonical momenta|canonical momentum]]. For a charged particle {{mvar|q}} in an [[electromagnetic field]], during a [[gauge transformation]], the position space [[wave function]] undergoes a [[Topological group|local]] [[U(1)]] group transformation,<ref>{{Cite journal|last1=Zinn-Justin|first1=Jean|last2=Guida|first2=Riccardo| date=2008-12-04|title=Gauge invariance|journal=Scholarpedia|language=en| volume=3|issue=12|pages=8287|doi=10.4249/scholarpedia.8287|bibcode=2008SchpJ...3.8287Z |issn=1941-6016|doi-access=free}}</ref> and <math display="inline"> \hat{p}\psi = - i \hbar \frac{\partial \psi}{\partial x} </math> will change its value. Therefore, the canonical momentum is not [[Gauge invariance|gauge invariant]], and hence not a measurable physical quantity.

The [[kinetic momentum]], a gauge invariant physical quantity, can be expressed in terms of the canonical momentum, the [[scalar potential]]&nbsp;{{mvar|φ}} and [[vector potential]]&nbsp;{{math|'''A'''}}:<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>
<math display="block">\mathbf{\hat{P}} = -i\hbar\nabla - q\mathbf{A} </math>

The expression above is called [[minimal coupling]]. For electrically neutral particles, the canonical momentum is equal to the kinetic momentum.

== Properties ==

=== Hermiticity ===
The momentum operator can be described as a [[symmetric operator|symmetric]] (i.e. Hermitian), [[unbounded operator]] acting on a dense subspace of the [[quantum state space]]. If the operator acts on a ([[Normalisable wave function|normalizable]]) [[quantum state]] then the operator is [[self-adjoint operator|self-adjoint]]. In physics the term '''Hermitian''' often refers to both symmetric and self-adjoint operators.<ref>See [http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf Lecture notes 1 by Robert Littlejohn] {{Webarchive|url=https://web.archive.org/web/20120617144946/http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf |date=2012-06-17 }} 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><ref>{{cite book| last=Hall | first=B. C. | title=Quantum Theory for Mathematicians | publisher=Springer |series=Graduate Texts in Mathematics | volume=267 | year=2013 | bibcode=2013qtm..book.....H | isbn=978-1461471158|page=64}}
</ref>

(In certain artificial situations, such as the quantum states on the semi-infinite interval {{closed-open|0, ∞}}, there is no way to make the momentum operator Hermitian.<ref>{{cite journal|author= Bonneau, G., Faraut, J., Valent, G.|title=Self-adjoint extensions of operators and the teaching of quantum mechanics|journal=American Journal of Physics |volume=69|pages=322–331| date=2001 | doi=10.1119/1.1328351 |issue=3 |arxiv=quant-ph/0103153|bibcode = 2001AmJPh..69..322B |s2cid=16949018 }}</ref> This is closely related to the fact that a semi-infinite interval cannot have translational symmetry—more specifically, it does not have [[unitary operator|unitary]] [[translation operator (quantum mechanics)|translation operators]]. See [[#Derivation from infinitesimal translations|below]].)

=== Canonical commutation relation ===
{{further information|Canonical commutation relation}}

By applying the [[Commutator#Ring theory|commutator]] to an arbitrary state in either the position or momentum basis, one can easily show that:
<math display="block"> \left [ \hat{x}, \hat{p} \right ] = \hat{x} \hat{p} - \hat{p} \hat{x} = i \hbar  \mathbb{I}, </math>
where <math> \mathbb{I}</math> is the [[identity function|unit operator]].<ref>{{cite journal | last1=Born | first1=M. | last2=Jordan | first2=P. | title=Zur Quantenmechanik | journal=Zeitschrift für Physik | volume=34 | issue=1 | date=1925 | issn=1434-6001 | doi=10.1007/BF01328531 | pages=858–888 | bibcode=1925ZPhy...34..858B | language=de}}</ref>
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]].

=== Fourier transform ===
The following discussion uses the [[bra–ket notation]].  One may write 
<math display="block"> \psi(x)=\langle x|\psi\rangle =\int\!\!dp~ \langle x|p\rangle \langle p|\psi\rangle = \int\!\!dp~ {e^{ixp/\hbar} \tilde \psi (p) \over \sqrt{2\pi\hbar}},</math> 
so the tilde represents the Fourier transform, in converting from coordinate space to momentum space.  It then holds that 
<math display="block">  \hat{p}=   \int\!\!dp~      |  p    \rangle p \langle p|=   -i\hbar      \int\!\!dx~ |  x \rangle  \frac{ d}{d x} \langle x| ~,  </math>
that is, the momentum acting in coordinate space corresponds to spatial frequency,
<math display="block"> \langle  x | \hat{p} | \psi \rangle = - i \hbar \frac{d}{dx} \psi ( x ) .</math>

An analogous result  applies for the position operator in the momentum basis,
<math display="block"> \langle  p | \hat{x} | \psi \rangle =  i \hbar \frac{d}{dp} \psi ( p ), </math>
leading to  further useful relations,
<math display="block"> \langle p | \hat{x} | p' \rangle = i \hbar \frac{d}{dp} \delta (p - p') ,</math>
<math display="block"> \langle x | \hat{p} | x' \rangle = -i \hbar \frac{d}{dx} \delta (x - x') ,</math>
where {{mvar|δ}} stands for [[Dirac's delta function]].

== Derivation from infinitesimal translations ==
{{see also|Noether's theorem|Stone's theorem on one-parameter unitary groups}}

The [[Translation operator (quantum mechanics)|translation operator]] is denoted {{math|''T''(''ε'')}}, where {{mvar|ε}} represents the length of the translation. It satisfies the following identity:
<math display="block"> T(\varepsilon) | \psi \rangle =  \int dx T(\varepsilon) | x \rangle \langle x | \psi \rangle </math>
that becomes
<math display="block">\int dx | x + \varepsilon \rangle \langle x |\psi \rangle = \int dx | x \rangle \langle x - \varepsilon | \psi \rangle = \int dx | x \rangle  \psi(x - \varepsilon) </math>

Assuming the function <math>\psi</math> to be [[analytic function|analytic]] (i.e. [[Differentiable function|differentiable]] in some domain of the [[complex plane]]), one may expand in a [[Taylor series]] about {{mvar|x}}:
<math display="block">\psi(x-\varepsilon) = \psi(x) - \varepsilon \frac{d \psi}{dx} </math>
so for [[infinitesimal]] values of {{mvar|ε}}:
<math display="block"> T(\varepsilon) = 1 - \varepsilon {d \over dx}  = 1 - {i \over \hbar} \varepsilon \left ( - i \hbar { d \over dx} \right )</math>

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:<ref>{{Cite book |last1=Sakurai |first1=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |location=Cambridge}}</ref>{{Explain|date=November 2022}}
<math display="block"> T(\varepsilon) =  1 - \frac{i}{\hbar} \varepsilon \hat{p}</math>
thus
<math display="block"> \hat{p} = - i \hbar \frac{d}{dx}. </math>

== 4-momentum operator ==
Inserting the 3d momentum operator above and the [[energy operator]] into the [[4-momentum]] (as a [[1-form]] with {{math|(+&thinsp;−&thinsp;−&thinsp;−)}} [[metric signature]]):
<math display="block">P_\mu = \left(\frac{E}{c},-\mathbf{p}\right)</math>
obtains the '''4-momentum operator''':
<math display="block">\hat{P}_\mu = \left(\frac{1}{c}\hat{E},-\mathbf{\hat{p}}\right) = i\hbar\left(\frac{1}{c} \frac{\partial}{\partial t}, \nabla\right) = i\hbar\partial_\mu </math>
where {{math|∂<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]].

The [[Dirac operator]] and [[Dirac slash]] of the 4-momentum is given by contracting with the [[gamma matrices]]:
<math display="block"> \gamma^\mu\hat{P}_\mu = i\hbar \gamma^\mu\partial_\mu = \hat{P} = i\hbar\partial \!\!\!/</math>

If the signature was {{math|(−&thinsp;+&thinsp;+&thinsp;+)}}, the operator would be
<math display="block">\hat{P}_\mu = \left(-\frac{1}{c}\hat{E},\mathbf{\hat{p}}\right) = -i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right) = -i\hbar\partial_\mu</math>
instead.

== See also ==
* [[Mathematical descriptions of the electromagnetic field]]
* [[Translation operator (quantum mechanics)]]
* [[Relativistic wave equations]]
* [[Pauli–Lubanski pseudovector]]

== References ==
{{reflist}}

{{Physics operator}}

[[Category:Quantum mechanics]]