When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ i\hbar\frac{d}{dt}\hat{U}=H\hat{U}. $$ And in this case, Hamiltonian in Heisenberg picture ($H_{H}$) is just the same thatas the Hamiltonian in Schrödinger picture ($H_{S}$), i.e. it commutes with $\hat{U}$. Now iI have a Hamiltonian that depensdepends explicitly on time. Specifically, $$H_{S}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega \hat{q}^2-F_0 \sin(\omega_0t)\hat{q}.$$
And in my problem I need to calculate $H_H$ (Hamiltonian in Heisenberg picture).
I've found that differential equation for $\hat{U}$ (I've mentioned about it above.) has generally solution in the form (with $U(0)=1$) $$U(t)=1+\xi\int\limits_0^t H(t')\,dt'+ \xi^2\int\limits_0^t H(t')\,dt'\int\limits_0^t' H(t'')\,dt''+\xi^3\int\limits_0^t H(t')\,dt'\int\limits_0^t' H(t'')\,dt''\int\limits_0^t'' H(t''')\,dt'''+...$$
So my questions are:
- Is there other ways to calculate $\hat{U}$, could give a link or tell me about them?
- If you know form of the solution for my case, please tell me.
- If you know any articles or papers articles on this topice, please link them to me, too.