I was just trying to confirm to myself that the following density operator

\rho(t) = exp(-i/hbar * H t) * \rho(0) * exp(i/hbar * H t)

fulfills the Liouville-von Neumann equation:

(d/dt (\rho(t)) = - i/hbar * [H,\rho(t)]).

, where [H,\rho(t)] denotes the commutator of the Hamiltonian with the density operator here. I only have to take the derivative of \rho(t) in order to plug it in, but although the answer is probably trivial, I am struggling at the moment as I don’t really know how to take the derivative of this expression (there are several operators acting here, so I am not sure how to apply the product rule here). Can someone please help me?

I was just trying to confirm to myself that the following density operator

$$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$$

fulfills the Liouville-von Neumann equation:

$$\frac{d}{dt}\rho(t) = - \frac{i}{\hbar} [H,\rho(t)]$$ where $[H,\rho(t)]$ denotes the commutator of the Hamiltonian with the density operator here. I only have to take the derivative of $\rho(t)$ in order to plug it in, but although the answer is probably trivial, I am struggling at the moment as I don’t really know how to take the derivative of this expression (there are several operators acting here, so I am not sure how to apply the product rule here). Can someone please help me?