Commutators

Topics

• Anticommutators

$[\hat{A},\hat{B}]\equiv \hat{A}\hat{B}-\hat{B}\hat{A}$

• Measures how badly operators commute, so if $=0$, then they commute

$[\hat{A}+\hat{B},\hat{C}]=[\hat{A},\hat{C}]+[\hat{B},\hat{C}]$

$[\hat{A}\hat{B},\hat{C}]=\hat{A}[\hat{B},\hat{C}]+[\hat{A},\hat{C}]\hat{B}$

Topics

• Canonical Commutation Relation

Examples

$[{\hat{a}}_{-},{\hat{a}}_{+}]=1$

$[\hat{x},V(x)]=0$

$[\hat{x},\hat{p}]=i\hslash$

$[\hat{x},{\hat{p}}^2]=2i\hslash p$

$[V(x),\hat{p}]=i\hslash \frac{\partial V}{\partial x}$

$[\hat{x},\hat{H}]=\frac{i\hslash \hat{p}}{m}$

3D Examples

$[r_i,p_j]=i\hslash {\delta }_{ij}$

• ${\delta }_{ij}$ is the Kronecker delta

$[r_i,r_j]=0$

$[p_i,p_j]=0$

$[L_i,L_j]=i\hslash {\epsilon }_{ijk}{L}_k$

• ${\epsilon }_{ijk}$ is the Levi-Civita symbol
• $1,2,3$ corresponds to $x,y,z$

$[L_z,L_{\pm }]=\pm \hslash L_{\pm }$

$[L^2,L_i]=0$

$[L_i,r_j]=i\hslash {\epsilon }_{ijk}{r}_k$

$[L_i,p_j]=i\hslash {\epsilon }_{ijk}{p}_k$

$[L_i,r]=0$

• $r=\sqrt{x^2+y^2+z^2}$

$[L_i,r^2]=0$

$[L_i,p^2]=0$

$[S_i,S_j]=i\hslash {\epsilon }_{ijk}{S}_k$