Use of Interaction Picture
The purpose of the interaction picture is to shunt all the time dependence due to H0 onto the operators, leaving only H1, I affecting the time-dependence of the state vectors.
The interaction picture is convenient when considering the effect of a small interaction term, H1, S, being added to the Hamiltonian of a solved system, H0, S. By switching into the interaction picture, you can use time-dependent perturbation theory to find the effect of H1, I.
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