Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 61 / 1 / d

Differentiate $U_I=U_0^{†}U$. The free propagator satisfies ${\dot{U}}_0^{†}=(i/\hslash )U_0^{†}{H}_0$, so the Schrödinger equation givesThus the interaction picture removes the drift and gives $\boxed{i\hslash {\dot{U}}_I=f(t)H_I(t)U_I,H_I=U_0^{†}{H}_1{U}_0}$, with $U_I(0)=I$. The transformed control Quantum Hamiltonian is time-dependent even though the original control matrix is constant.