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

Question

In one of my exercise, I got following differential equation for density matrix $rho$,
$$
frac{drho}{dt}=-i[H_1,rho]+{H_2,rho}
$$
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.

Roger Vadim · Answer

The first step would be to write down the commutator and the anticommutator explicitly:
$$frac{drho}{dt} = (-iH_1 + H_2)rho + rho(iH_1+H_2).$$
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).

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

One Answer

Add your own answers!

Ask a Question