Commutators
#Physics
Topics
$\displaystyle [\hat{A},\hat{B}]\equiv\hat{A}\hat{B}-\hat{B}\hat{A}$
- Measures how badly operators commute, so if $\displaystyle =0$, then they commute
$\displaystyle [\hat{A}+\hat{B},\hat{C}]=[\hat{A},\hat{C}]+[\hat{B},\hat{C}]$
$\displaystyle [\hat{A}\hat{B},\hat{C}]=\hat{A}[\hat{B},\hat{C}]+[\hat{A},\hat{C}]\hat{B}$
Topics
Examples
$\displaystyle [\hat{a}{-},\hat{a}{+}]=1$
$\displaystyle [\hat{x},V(x)]=0$
$\displaystyle [\hat{x},\hat{p}]=i\hbar$
$\displaystyle [\hat{x},\hat{p}^{2}]=2i\hbar p$
$\displaystyle [V(x),\hat{p}]=i\hbar \frac{ \partial V }{ \partial x }$
$\displaystyle [\hat{x},\hat{H}]=\frac{i\hbar \hat{p}}{m}$
3D Examples
$\displaystyle [r_{i},p_{j}]=i\hbar \delta_{ij}$
- $\displaystyle \delta_{ij}$ is the Kronecker delta
$\displaystyle [r_{i},r_{j}]=0$
$\displaystyle [p_{i},p_{j}]=0$
$\displaystyle [L_{i},L_{j}]=i\hbar \varepsilon_{ijk}L_{k}$
- $\displaystyle \varepsilon_{ijk}$ is the Levi-Civita symbol
- $\displaystyle 1,2,3$ corresponds to $\displaystyle x,y,z$
$\displaystyle [L_{z},L_{\pm }]=\pm \hbar L_{\pm }$
$\displaystyle [L^{2},L_{i}]=0$
$\displaystyle [L_{i},r_{j}]=i\hbar \varepsilon_{ijk}r_{k}$
$\displaystyle [L_{i},p_{j}]=i\hbar \varepsilon_{ijk}p_{k}$
$\displaystyle [L_{i},r]=0$
- $\displaystyle r=\sqrt{ x^{2}+y^{2}+z^{2} }$