Elementary Commutators

We need $[p_x,\cos \frac{\pi x}{L}]$ and $[p_x,\sin\frac{\pi x}{L}]$:

$$[p_x,\cos \frac{\pi x}{L}]= \imath \hbar \frac{\pi}{L} \sin \... ...\right ) = p_x \cos \frac{\pi x}{L} - \cos \frac{\pi x}{L} p_x$$

$$[p_x,\sin \left ( \frac{\pi x}{L} \right )]= -\imath \hbar \f... ...\pi x}{L} \right ) - \sin \left ( \frac{\pi x}{L} \right ) p_x$$

and $[p_x^2,\cos\left ( \frac{\pi x}{L} \right )]$ (and its sine equivalent):

$$[p_x^2,\cos \left ( \frac{\pi x}{L} \right )] = 2 \imath \hba... ...\frac{\pi}{L} \right )^2 \cos \left ( \frac{\pi x}{L} \right )$$

$$[p_x^2,\sin \left ( \frac{\pi x}{L} \right )] = - 2 \imath \h... ...(\frac{\pi}{L}\right )^2 \sin \left ( \frac{\pi x}{L} \right )$$