Editing Operator (physics) (section)

==Operators in quantum mechanics==

The [[mathematical formulation of quantum mechanics]] (QM) is built upon the concept of an operator.

Physical [[pure state]]s in quantum mechanics are represented as [[unit-norm vector]]s (probabilities are normalized to one) in a special [[complex number|complex]] [[Hilbert space]]. [[Time evolution]] in this [[vector space]] is given by the application of the [[evolution operator]].

Any [[observable]], i.e., any quantity which can be measured in a physical experiment, should be associated with a [[self-adjoint]] [[linear operator]]. The operators must yield real [[eigenvalue]]s, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be [[Hermitian matrix|Hermitian]].<ref name="QUANTUM CHEMISRTY 1977">Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P.W. Atkins, Oxford University Press, 1977, {{ISBN|0-19-855129-0}}</ref> The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue. See below for mathematical details about Hermitian operators.

In the [[Schrödinger equation#Particles as waves|wave mechanics]] formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see [[position and momentum space]] for details), so observables are [[differential operator]]s.

In the [[matrix mechanics]] formulation, the [[Norm (mathematics)|norm]] of the physical state should stay fixed, so the evolution operator should be [[unitary transformation|unitary]], and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction.

===Wavefunction ===

{{Main|wavefunction}}

The wavefunction must be [[square-integrable]] (see [[Lp space|''L<sup>p</sup>'' spaces]]), meaning:

:<math>\iiint_{\R^3} |\psi(\mathbf{r})|^2 \, d^3\mathbf{r} = \iiint_{\R^3} \psi(\mathbf{r})^*\psi(\mathbf{r}) \, d^3\mathbf{r} < \infty </math>

and normalizable, so that:

:<math>\iiint_{\R^3} |\psi(\mathbf{r})|^2 \, d^3\mathbf{r} = 1 </math>

Two cases of eigenstates (and eigenvalues) are:
* for '''discrete''' eigenstates <math> | \psi_i \rangle </math> forming a discrete basis, so any state is a [[summation|sum]] <math display="block">|\psi\rangle = \sum_i c_i|\phi_i\rangle</math> where ''c<sub>i</sub>'' are complex numbers such that {{!}}''c<sub>i</sub>''{{!}}<sup>2</sup> = ''c<sub>i</sub>''<sup>*</sup>''c<sub>i</sub>'' is the probability of measuring the state <math>|\phi_i\rangle</math>, and the corresponding set of eigenvalues ''a<sub>i</sub>'' is also discrete - either [[Finite set|finite]] or [[countably infinite]]. In this case, the inner product of two eigenstates is given by <math>\langle \phi_i \vert \phi_j\rangle=\delta_{ij}</math>, where <math>\delta_{mn}</math> denotes the [[Kronecker delta|Kronecker Delta]]. However, 
* for a '''continuum''' of eigenstates forming a continuous basis, any state is an [[integral]] <math display="block">|\psi\rangle = \int c(\phi) \, d\phi|\phi\rangle </math> where ''c''(''φ'') is a complex function such that {{!}}''c''(φ){{!}}<sup>2</sup> = ''c''(φ)<sup>*</sup>''c''(φ) is the probability of measuring the state <math>|\phi\rangle</math>, and there is an [[uncountably infinite]] set of eigenvalues ''a''. In this case, the inner product of two eigenstates is defined as <math>\langle \phi' \vert \phi\rangle=\delta(\phi - \phi')</math>, where here <math>\delta(x-y)</math> denotes the [[Dirac delta|Dirac Delta]].

===Linear operators in wave mechanics===

{{Main|Wave function|Bra–ket notation}}

Let {{math|''ψ''}} be the wavefunction for a quantum system, and <math>\hat{A}</math> be any [[linear operator]] for some observable {{math|''A''}} (such as position, momentum, energy, angular momentum etc.). If {{math|''ψ''}} is an eigenfunction of the operator <math>\hat{A}</math>, then

:<math>\hat{A} \psi = a \psi ,</math>

where {{math|''a''}} is the [[Eigenvalues and eigenvectors|eigenvalue]] of the operator, corresponding to the measured value of the observable, i.e. observable {{math|''A''}} has a measured value {{math|''a''}}.

If {{math|''ψ''}}  is an eigenfunction of a given operator <math>\hat{A}</math>, then a definite quantity (the eigenvalue {{math|''a''}}) will be observed if a measurement of the observable {{math|''A''}} is made on the state {{math|''ψ''}}. Conversely, if {{math|''ψ''}} is not an eigenfunction of <math>\hat{A}</math>, then it has no eigenvalue for <math>\hat{A}</math>, and the observable does not have a single definite value in that case. Instead, measurements of the observable {{math|''A''}} will yield each eigenvalue with a certain probability (related to the decomposition of {{math|''ψ''}} relative to the orthonormal eigenbasis of <math>\hat{A}</math>).

In bra–ket notation the above can be written;

:<math>\begin{align}
  \hat{A} \psi &= \hat{A} \psi ( \mathbf{r} ) = \hat{A} \left\langle \mathbf{r} \mid \psi \right\rangle = \left\langle \mathbf{r} \left\vert \hat {A} \right\vert \psi \right\rangle \\
        a \psi &= a \psi ( \mathbf{r} ) = a \left\langle \mathbf{r} \mid \psi \right\rangle = \left\langle \mathbf{r} \mid a \mid \psi \right\rangle \\
\end{align} </math>

that are equal if <math> \left| \psi \right\rangle </math> is an [[eigenvector]], or [[eigenket]] of the observable {{math|''A''}}.

Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the [[del operator]], which is itself a vector (useful in momentum-related quantum operators, in the table below).

An operator in ''n''-dimensional space can be written:

:<math> \mathbf{\hat{A}} = \sum_{j=1}^n \mathbf{e}_j \hat{A}_j </math>

where '''e'''<sub>''j''</sub> are basis vectors corresponding to each component operator ''A<sub>j</sub>''. Each component will yield a corresponding eigenvalue <math>a_j</math>. Acting this on the wave function {{math|''ψ''}}:

:<math> \mathbf{\hat{A}} \psi = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \psi = \sum_{j=1}^n \left( \mathbf{e}_j \hat{A}_j \psi \right) = \sum_{j=1}^n \left( \mathbf{e}_j a_j \psi \right) </math>

in which we have used <math> \hat{A}_j \psi = a_j \psi .</math>

In bra–ket notation:

:<math>\begin{align}
  \mathbf{\hat{A}} \psi = \mathbf{\hat{A}} \psi ( \mathbf{r} ) = \mathbf{\hat{A}} \left\langle \mathbf{r} \mid \psi \right\rangle
    &= \left\langle \mathbf{r} \left\vert \mathbf{\hat{A}} \right\vert \psi \right\rangle \\
  \left ( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right ) \psi = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \psi ( \mathbf{r} ) = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \left\langle \mathbf{r} \mid \psi \right\rangle
    &= \left\langle \mathbf{r} \left\vert \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right\vert \psi \right\rangle
\end{align}</math>

===Commutation of operators on ''Ψ''===

{{main|Commutator}}

If two observables ''A'' and ''B'' have linear operators <math> \hat{A} </math> and <math> \hat{B} </math>, the commutator is defined by,

:<math> \left[ \hat{A}, \hat{B} \right] = \hat{A} \hat{B} - \hat{B} \hat{A} </math>

The commutator is itself a (composite) operator. Acting the commutator on ''ψ'' gives:

:<math> \left[ \hat{A}, \hat{B} \right] \psi = \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi . </math>

If ''ψ'' is an eigenfunction with eigenvalues ''a'' and ''b'' for observables ''A'' and ''B'' respectively, and if the operators commute:

:<math> \left[ \hat{A}, \hat{B} \right] \psi = 0, </math>

then the observables ''A'' and ''B'' can be measured simultaneously with infinite precision, i.e., uncertainties <math> \Delta A = 0 </math>, <math> \Delta B = 0 </math> simultaneously. ''ψ'' is then said to be the simultaneous eigenfunction of A and B. To illustrate this:

:<math> \begin{align}
  \left[ \hat{A}, \hat{B} \right] \psi &= \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi \\
    & = a(b \psi) - b(a \psi) \\
    & = 0 . \\
\end{align} </math>

It shows that measurement of A and B does not cause any shift of state, i.e., initial and final states are same (no disturbance due to measurement). Suppose we measure A to get value a. We then measure B to get the value b. We measure A again. We still get the same value a. Clearly the state (''ψ'') of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision.

If the operators do not commute:

:<math> \left[ \hat{A}, \hat{B} \right] \psi \neq 0, </math>

they cannot be prepared simultaneously to arbitrary precision, and there is an [[uncertainty relation]] between the observables

:<math>\Delta A \Delta B \geq \left|\frac{1}{2}\langle[A, B]\rangle\right|</math> 
even if ''ψ''  is an eigenfunction the above relation holds. Notable pairs are position-and-momentum and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as ''L<sub>x</sub>'' and ''L<sub>y</sub>'', or ''s<sub>y</sub>'' and ''s<sub>z</sub>'', etc.).<ref name=Ballentine1970>{{citation
 | last =Ballentine | first =L. E.
 | title =The Statistical Interpretation of Quantum Mechanics
 | journal =Reviews of Modern Physics 
 | volume =42
 | issue =4
 | pages =358–381
 | year =1970
 | doi =10.1103/RevModPhys.42.358
 | bibcode = 1970RvMP...42..358B
}}</ref>

===Expectation values of operators on ''Ψ''===

The [[Expectation value (quantum mechanics)|expectation value]] (equivalently the average or mean value) is the average measurement of an observable, for particle in region ''R''. The expectation value <math>\left\langle \hat{A} \right\rangle </math> of the operator <math> \hat{A} </math> is calculated from:<ref name="Quantum Mechanics Demystified 2006">Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, {{ISBN|0-07-145546-9}}</ref>

:<math>\left\langle \hat{A} \right\rangle = \int_R \psi^{*}\left( \mathbf{r} \right) \hat{A} \psi \left( \mathbf{r} \right) \mathrm{d}^3\mathbf{r} = \left\langle \psi \left| \hat{A} \right| \psi \right\rangle .</math>

This can be generalized to any function ''F'' of an operator:

:<math> \left\langle F \left( \hat{A} \right) \right\rangle = \int_R \psi(\mathbf{r})^{*} \left[ F \left( \hat{A} \right) \psi(\mathbf{r}) \right] \mathrm{d}^3 \mathbf{r} = \left\langle \psi \left| F \left( \hat{A} \right) \right| \psi \right\rangle , </math>

An example of ''F'' is the 2-fold action of ''A'' on ''ψ'', i.e. squaring an operator or doing it twice:

:<math>\begin{align}
                             F\left(\hat{A}\right) &= \hat{A}^2 \\
  \Rightarrow \left\langle \hat{A}^2 \right\rangle &= \int_R \psi^{*} \left( \mathbf{r} \right) \hat{A}^2 \psi \left( \mathbf{r} \right) \mathrm{d}^3\mathbf{r} = \left\langle \psi \left\vert \hat{A}^2 \right\vert \psi \right\rangle \\
\end{align}\,\!</math>

===Hermitian operators===

{{Main|Self-adjoint operator}}

The definition of a [[Hermitian operator]] is:<ref name="QUANTUM CHEMISRTY 1977"/>

:<math>\hat{A} = \hat{A}^\dagger</math>

Following from this, in bra–ket notation:

:<math>\left\langle \phi_i \left| \hat{A} \right| \phi_j \right\rangle = \left\langle \phi_j \left| \hat{A} \right| \phi_i \right\rangle^*.</math>

Important properties of Hermitian operators include:
* real eigenvalues,
* eigenvectors with different eigenvalues are [[orthogonal]],
* eigenvectors can be chosen to be a complete [[orthonormal basis]],

===Operators in matrix mechanics ===

An operator can be written in matrix form to map one basis vector to another. Since the operators are linear, the matrix is a [[linear transformation]] (aka transition matrix) between bases. Each basis element <math>\phi_j </math> can be connected to another,<ref name="Quantum Mechanics Demystified 2006"/> by the expression:

:<math>A_{ij} = \left\langle \phi_i \left| \hat{A} \right| \phi_j \right\rangle,</math>

which is a matrix element:

:<math>\hat{A} = \begin{pmatrix}
  A_{11} & A_{12} & \cdots & A_{1n} \\
  A_{21} & A_{22} & \cdots & A_{2n} \\
  \vdots & \vdots & \ddots & \vdots \\
  A_{n1} & A_{n2} & \cdots & A_{nn} \\
\end{pmatrix}
</math>

A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.<ref name="QUANTUM CHEMISRTY 1977"/> In matrix form, operators allow real eigenvalues to be found, corresponding to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the [[characteristic polynomial]]:

:<math> \det\left( \hat{A} - a \hat{I} \right) = 0 ,</math>

where ''I'' is the ''n'' × ''n'' [[identity matrix]], as an operator it corresponds to the identity operator. For a discrete basis:

:<math> \hat{I} = \sum_i |\phi_i\rangle\langle\phi_i|</math>

while for a continuous basis:

:<math> \hat{I} = \int |\phi\rangle\langle\phi| \mathrm{d}\phi</math>

=== Inverse of an operator ===

A non-singular operator <math>\hat{A}</math> has an inverse <math> \hat{A}^{-1} </math> defined by:

:<math> \hat{A}\hat{A}^{-1} = \hat{A}^{-1}\hat{A} = \hat{I} </math>

If an operator has no inverse, it is a singular operator. In a finite-dimensional space, an operator is non-singular if and only if its determinant is nonzero:

:<math> \det\left(\hat{A}\right) \neq 0</math>

and hence the determinant is zero for a singular operator.

===Table of Quantum Mechanics operators ===

The operators used in quantum mechanics are collected in the table below (see for example<ref name="QUANTUM CHEMISRTY 1977"/><ref>[https://feynmanlectures.caltech.edu/III_20.html Operators - The Feynman Lectures on Physics]</ref>). The bold-face vectors with circumflexes are not [[unit vector]]s, they are 3-vector operators; all three spatial components taken together.

:{| class="wikitable"
|- style="vertical-align:top;"
! scope="col" | Operator (common name/s)
! scope="col" | Cartesian component
! scope="col" | General definition
! scope="col" | SI unit
! scope="col" | Dimension
|- style="vertical-align:top;"
! [[Position operator|Position]]
| <math>\begin{align}
  \hat{x} &= x, &
  \hat{y} &= y, &
  \hat{z} &= z 
\end{align}</math>
| <math> \mathbf{\hat{r}} = \mathbf{r} \,\!</math>
| m
| [L]
|- style="vertical-align:top;"
!rowspan="2"| [[Momentum operator|Momentum]]
| General
<math> \begin{align}
  \hat{p}_x & = -i \hbar \frac{\partial}{\partial x}, &
  \hat{p}_y & = -i \hbar \frac{\partial}{\partial y}, &
  \hat{p}_z & = -i \hbar \frac{\partial}{\partial z} 
\end{align}</math> 
| General
<math> \mathbf{\hat{p}} = -i \hbar \nabla \,\!</math>
| J s m<sup>−1</sup> = N s
| [M] [L] [T]<sup>−1</sup>
|- style="vertical-align:top;"
| Electromagnetic field
<math> \begin{align}
  \hat{p}_x = -i \hbar \frac{\partial}{\partial x} - qA_x \\
  \hat{p}_y = -i \hbar \frac{\partial}{\partial y} - qA_y \\
  \hat{p}_z = -i \hbar \frac{\partial}{\partial z} - qA_z 
\end{align}</math>
| Electromagnetic field (uses [[kinetic momentum]]; '''A''', vector potential)
<math> \begin{align}
  \mathbf{\hat{p}} & = \mathbf{\hat{P}} - q\mathbf{A} \\
                   & = -i \hbar \nabla - q\mathbf{A} \\
\end{align}\,\!</math>
| J s m<sup>−1</sup> = N s
| [M] [L] [T]<sup>−1</sup>
|- style="vertical-align:top;"
!rowspan="3"| [[Kinetic energy]]
| Translation
<math> \begin{align}
  \hat{T}_x & = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \\[2pt]
  \hat{T}_y & = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial y^2} \\[2pt]
  \hat{T}_z & = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2} \\
\end{align} </math> 
|
<math> \begin{align}
  \hat{T} & = \frac{1}{2m}\mathbf{\hat{p}}\cdot\mathbf{\hat{p}} \\
          & = \frac{1}{2m}(-i \hbar \nabla)\cdot(-i \hbar \nabla) \\
          & = \frac{-\hbar^2 }{2m}\nabla^2
\end{align}\,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|- style="vertical-align:top;"
| Electromagnetic field
<math> \begin{align}
  \hat{T}_x & = \frac{1}{2m}\left(-i \hbar \frac{\partial}{\partial x} - q A_x \right)^2 \\
  \hat{T}_y & = \frac{1}{2m}\left(-i \hbar \frac{\partial}{\partial y} - q A_y \right)^2 \\
  \hat{T}_z & = \frac{1}{2m}\left(-i \hbar \frac{\partial}{\partial z} - q A_z \right)^2 
\end{align}\,\!</math>
| Electromagnetic field ('''A''', [[vector potential]])
<math> \begin{align}
  \hat{T} & = \frac{1}{2m}\mathbf{\hat{p}}\cdot\mathbf{\hat{p}} \\
          & = \frac{1}{2m}(-i \hbar \nabla - q\mathbf{A})\cdot(-i \hbar \nabla - q\mathbf{A}) \\
          & = \frac{1}{2m}(-i \hbar \nabla - q\mathbf{A})^2
\end{align}\,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|- style="vertical-align:top;"
| Rotation (''I'', [[moment of inertia]])
<math> \begin{align} 
  \hat{T}_{xx} & = \frac{\hat{J}_x^2}{2I_{xx}} \\
  \hat{T}_{yy} & = \frac{\hat{J}_y^2}{2I_{yy}} \\
  \hat{T}_{zz} & = \frac{\hat{J}_z^2}{2I_{zz}} \\
\end{align}\,\!</math>
| Rotation
<math> \hat{T} = \frac{\mathbf{\hat{J}}\cdot\mathbf{\hat{J}}}{2I} \,\!</math>{{Citation needed|reason=does not seem to be correct, inconsistent with the Cartesian entriesdate=November 2012|date=November 2012}}
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|- style="vertical-align:top;"
! Potential energy
| N/A
|<math> \hat{V} = V\left( \mathbf{r}, t \right) = V \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|- style="vertical-align:top;"
! Total [[energy operator|energy]]
| N/A
| Time-dependent potential:<br />
<math> \hat{E} = i \hbar \frac{\partial}{\partial t} \,\!</math>

Time-independent:<br />
<math> \hat{E} = E \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|- style="vertical-align:top;"
! [[Hamiltonian operator|Hamiltonian]]
|
|<math> \begin{align}
  \hat{H} & = \hat{T} + \hat{V} \\
          & = \frac{1}{2m}\mathbf{\hat{p}}\cdot\mathbf{\hat{p}} + V \\
          & = \frac{1}{2m}\hat{p}^2 + V \\
\end{align} \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|- style="vertical-align:top;"
! [[Angular momentum operator]]
| <math>\begin{align}
  \hat{L}_x & = -i\hbar \left(y {\partial \over \partial z} - z {\partial \over \partial y}\right) \\
  \hat{L}_y & = -i\hbar \left(z {\partial \over \partial x} - x {\partial \over \partial z}\right) \\
  \hat{L}_z & = -i\hbar \left(x {\partial \over \partial y} - y {\partial \over \partial x}\right)
\end{align}</math>
| <math>\mathbf{\hat{L}} =  \mathbf{r} \times -i\hbar \nabla </math>
| J s = N s m
| [M] [L]<sup>2</sup> [T]<sup>−1</sup>
|- style="vertical-align:top;"
! [[Spin (physics)|Spin]] angular momentum
| <math>\begin{align}
  \hat{S}_x &= {\hbar \over 2} \sigma_x &
  \hat{S}_y &= {\hbar \over 2} \sigma_y &
  \hat{S}_z &= {\hbar \over 2} \sigma_z 
\end{align}</math>

where

<math>\begin{align}
  \sigma_x &= \begin{pmatrix}
    0 & 1 \\
    1 & 0
  \end{pmatrix} \\
  \sigma_y &= \begin{pmatrix}
    0 & -i \\
    i &  0
  \end{pmatrix} \\
  \sigma_z &= \begin{pmatrix}
    1 &  0 \\
    0 & -1
  \end{pmatrix}
\end{align}</math>

are the [[Pauli matrices]] for [[spin-1/2]] particles.
| <math>\mathbf{\hat{S}} = {\hbar \over 2} \boldsymbol{\sigma} \,\!</math>

where '''σ''' is the vector whose components are the Pauli matrices.
| J s = N s m
| [M] [L]<sup>2</sup> [T]<sup>−1</sup>
|- style="vertical-align:top;"
! Total angular momentum
| <math>\begin{align}
  \hat{J}_x & = \hat{L}_x + \hat{S}_x \\
  \hat{J}_y & = \hat{L}_y + \hat{S}_y \\
  \hat{J}_z & = \hat{L}_z + \hat{S}_z
\end{align}</math>
| <math>\begin{align}
  \mathbf{\hat{J}} & = \mathbf{\hat{L}} + \mathbf{\hat{S}} \\
                   & = -i\hbar \mathbf{r}\times\nabla + \frac{\hbar}{2}\boldsymbol{\sigma} 
\end{align}</math>
| J s = N s m
| [M] [L]<sup>2</sup> [T]<sup>−1</sup>
|- style="vertical-align:top;"
! [[Transition dipole moment]] (electric)
| <math>\begin{align}
  \hat{d}_x & = q\hat{x}, &
  \hat{d}_y & = q\hat{y}, &
  \hat{d}_z & = q\hat{z}
\end{align}</math>
| <math>\mathbf{\hat{d}} = q \mathbf{\hat{r}} </math>
| C m
| [I] [T] [L]
|}

===Examples of applying quantum operators===

The procedure for extracting information from a wave function is as follows. Consider the momentum ''p'' of a particle as an example. The momentum operator in position basis in one dimension is:

:<math>\hat{p} = -i\hbar\frac{\partial }{\partial x}</math>

Letting this act on ''ψ'' we obtain:

:<math>\hat{p} \psi = -i\hbar\frac{\partial }{\partial x} \psi ,</math>

if ''ψ'' is an eigenfunction of <math>\hat{p}</math>, then the momentum eigenvalue ''p'' is the value of the particle's momentum, found by:

:<math> -i\hbar\frac{\partial }{\partial x} \psi = p \psi.</math>

For three dimensions the momentum operator uses the [[nabla symbol|nabla]] operator to become:

:<math>\mathbf{\hat{p}} = -i\hbar\nabla .</math>

In Cartesian coordinates (using the standard Cartesian basis vectors '''e'''<sub>x</sub>, '''e'''<sub>y</sub>, '''e'''<sub>z</sub>) this can be written;

:<math>\mathbf{e}_\mathrm{x}\hat{p}_x + \mathbf{e}_\mathrm{y}\hat{p}_y + \mathbf{e}_\mathrm{z}\hat{p}_z = -i\hbar\left ( \mathbf{e}_\mathrm{x} \frac{\partial }{\partial x} + \mathbf{e}_\mathrm{y} \frac{\partial }{\partial y} + \mathbf{e}_\mathrm{z} \frac{\partial }{\partial z} \right ),</math>

that is:

:<math> \hat{p}_x = -i\hbar \frac{\partial}{\partial x}, \quad \hat{p}_y = -i\hbar \frac{\partial}{\partial y} , \quad \hat{p}_z = -i\hbar \frac{\partial}{\partial z} \,\!</math>

The process of finding eigenvalues is the same. Since this is a vector and operator equation, if ''ψ'' is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum. Acting <math> \mathbf{\hat{p}} </math> on ''ψ'' obtains:

:<math> \begin{align}
\hat{p}_x \psi & = -i\hbar \frac{\partial}{\partial x} \psi = p_x \psi \\
\hat{p}_y \psi & = -i\hbar \frac{\partial}{\partial y} \psi = p_y \psi \\
\hat{p}_z \psi & = -i\hbar \frac{\partial}{\partial z} \psi = p_z \psi \\
\end{align} \,\!</math>