# How to solve differential equation involving commutator and anti-commutator?

In one of my exercise, I got following differential equation for density matrix $$\rho$$, $$\frac{d\rho}{dt}=-i[H_1,\rho]+\{H_2,\rho\}$$ where $$H_1$$ and $$H_2$$ are the Hermitian Hamiltonian, and $$[.,.]$$ is commutator, and $$\{.,.\}$$ is anti-commutator. I know that the commutator part on the r.h.s. causes rotation to our state. I have no idea what the anti-commutator part will do. Is there any method to solve such kind of equations? Can I make some statements about the solution of $$\rho$$, for e.g., how will it behave (maybe at short time or at long time), etc? Any help would be appreciated.

The first step would be to write down the commutator and the anticommutator explicitly: $$\frac{d\rho}{dt} = (-iH_1 + H_2)\rho + \rho(iH_1+H_2).$$ Now one could already guess the solution or solve it using any of the available methods for linear matrix equations. For example, the formal solution to $$\frac{dx}{dt} = Ax(t),$$ is $$x=x(0)\exp{(At)},$$ we can now try to use variation of constant by experimenting with $$\rho(t) = e^{-iH_1 t +H_2t}\tilde{\rho}(t)$$ and so on (being careful to preserve the order of the matrix exponents in respect to the density matrix).

• Thanks. What do you mean by "variation of constant"? Jun 3 '20 at 13:01