Interaction Picture, Rorating frame

Hamiltonian in the interaction picture

\[i\partial |\psi \rangle =H|\psi \rangle ,e^{iH_0t}|\psi \rangle =|\phi \rangle \\ i\partial \left( e^{-iH_0t}|\phi \rangle \right) =H\left( e^{-iH_0t}|\phi \rangle \right) \\ i\left( -iH_0e^{-iH_0t}|\phi \rangle +e^{-iH_0t}\partial |\phi \rangle \right) =H\left( e^{-iH_0t}|\phi \rangle \right) \\ H_0e^{-iH_0t}|\phi \rangle +ie^{-iH_0t}\partial |\phi \rangle =H\left( e^{-iH_0t}|\phi \rangle \right) \\ ie^{iH_0t}\partial |\phi \rangle =H\left( e^{-iH_0t}|\phi \rangle \right) -H_0e^{-iH_0t}|\phi \rangle \\ ie^{-iH_0t}\partial |\phi \rangle =\left( H-H_0 \right) e^{-iH_0t}|\phi \rangle \\ i\partial |\phi \rangle =e^{iH_0t}\left( H-H_0 \right) e^{-iH_0t}|\phi \rangle \]

So the hamiltonian in the interaction picture is first dropping the term \(H_0\) and then acting the unitary operator in both sides of the residual hamiltonian.

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Further reading