Introducció

Definició

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

Propietats més elementals

$$ \begin{aligned} [\hat{A}, \hat{B}] &=-[\hat{B}, \hat{A}] \\ [\hat{A}, c]&=0 \\ [\hat{A}, \hat{B}+\hat{C}] &= [\hat{A}, \hat{B}]+[\hat{A},\hat{C}] \\ [\hat{A}, \hat{B} \hat{C}] &=[\hat{A}, \hat{B}] \hat{C}+\hat{B}[\hat{A}, \hat{C}] \\ [\hat{A} \hat{B}, \hat{C}] &=[\hat{A}, \hat{C}] \hat{B}+\hat{A}[\hat{B}, \hat{C}] \\ [f(\hat{A}), \hat{A}]&=0 \end{aligned} $$

Identitat de Jacobi

$$ [\hat{A},[\hat{B}, \hat{C}]]+[\hat{B},[\hat{C}, \hat{A}]]+[\hat{C},[\hat{A}, \hat{B}]]=0 $$

Altres propietats

$$ [\hat{A},k\mathbb{I}]=0 $$

$$ e^{(A+B)}=e^A e^B e^{-\frac{1}{2}[A, B]} $$

Propietats de casos particulars

Si $A(s)=e^{sB}Ae^{-sB}$

$$ \frac{d A(s)}{ds}=[B,A(s)] $$

$$ A(s)=A+s[B, A]+\frac{s^2}{2}[B,[B, A]]+\frac{s^3}{3 !}[B,[B,[B, A]]]+\cdots $$

Si $[A,[A,B]]=0$

$$ [\hat{A}^n,\hat{B}]=[\hat{A},\hat{B}]n\hat{A}^{n-1} $$

$$ [f(\hat{A}),\hat{B}]=[\hat{A},\hat{B}]f'(\hat{A}) $$

$$ e^{t A} B e^{-t A}=B+t[A, B] $$

Extra