PX262 - C6a - commutation relations

in classical physics, the order of dynamical variables does not matter, as they commute
this is not the case in quantum mechanics
consider the position and momentum operators:

\begin{aligned} ({\hat{P}}_{x} \hat{X} - \hat{X} {\hat{P}}_{x}) ψ & = - i ℏ \frac{\partial}{\partial x} (x ψ) - x (- i ℏ \frac{\partial ψ}{\partial x}) \\ = - i ℏ (ψ + x \frac{\partial ψ}{\partial x}) + i ℏ x \frac{\partial ψ}{\partial x} \\ = - i ℏ ψ \end{aligned}

[{\hat{P}}_{x}, \hat{X}] = {\hat{P}}_{x} \hat{X} - \hat{X} {\hat{P}}_{x} = - i ℏ

\begin{matrix} [{\hat{X}}_{i}, {\hat{X}}_{j}] = [{\hat{P}}_{i}, {\hat{P}}_{j}] = 0 \\ [{\hat{P}}_{i}, {\hat{X}}_{j}] = - [{\hat{X}}_{j}, {\hat{P}}_{i}] = - i ℏ δ_{i j} \end{matrix}

note: these commutators are fundamental in quantum mechanics
the form of the operators, ${\hat{P}}_{i}$ , $\hat{X}$ , are not unique
the representation used here is called positional representation
a few useful identities:
$[\hat{A}, \hat{B}] = - [\hat{B}, \hat{A}]$
$[\hat{A}, \hat{B} \hat{C}] = [\hat{A}, \hat{B}] + [\hat{A}, \hat{C}]$
$[\hat{A}, \hat{B}, \hat{C}] = [\hat{A}, \hat{B}] \hat{C} + \hat{B} [\hat{A}, \hat{C}]$
$\hat{A} \hat{B} \hat{C} - \hat{B} \hat{C} \hat{A} + \hat{B} \hat{A} \hat{C} - \hat{B} \hat{A} \hat{C} = \hat{B} [\hat{A}, \hat{C}] + [\hat{A}, \hat{B}] \hat{C}$