Editing Operator (physics) (section)

===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]
|}