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.

And also we can do this for density matrix. Suppose we have Hamiltonian $H\left( t \right) =H_0\left( t \right) +V\left( t \right) $, then

\[\dot{\rho}\left( t \right) =-i\left[ H_0\left( t \right) +V\left( t \right) ,\rho \left( t \right) \right] \\ now\,\,define\,\,interaction\,\,picture\,\,w.r.t. H_0\left( t \right) \\ we\,\,have\,\,\tilde{\rho}\left( t \right) =e^{iH_0t}\rho \left( t \right) e^{-iH_0t} \\ by\,\,this \\ \dot{\tilde{\rho}}\left( t \right) =e^{iH_0t}iH_0\rho \left( t \right) e^{-iH_0t}-e^{iH_0t}\rho \left( t \right) iH_0e^{-iH_0t}+e^{iH_0t}\dot{\rho}\left( t \right) e^{-iH_0t} \\ =e^{iH_0t}iH_0\rho \left( t \right) e^{-iH_0t}-e^{iH_0t}\rho \left( t \right) iH_0e^{-iH_0t}-ie^{iH_0t}\left[ H_0\left( t \right) +V\left( t \right) ,\rho \left( t \right) \right] e^{-iH_0t} \\ in\,\,interaction\,\,picture\,\,w.r.t. H_0\left( t \right) ,every\,\,observable\,\,will\,\,be\,\,\tilde{A}=e^{iH_0t}Ae^{-iH_0t} \\ so\,\,\tilde{V}\left( t \right) =e^{iH_0t}V\left( t \right) e^{-iH_0t} \\ \dot{\tilde{\rho}}\left( t \right) =e^{iH_0t}iH_0\rho \left( t \right) e^{-iH_0t}-e^{iH_0t}\rho \left( t \right) iH_0e^{-iH_0t}-i\left[ H_0\left( t \right) +\tilde{V}\left( t \right) ,\tilde{\rho}\left( t \right) \right] \\ =-i\left[ \tilde{\rho}\left( t \right) ,H_0 \right] -i\left[ H_0\left( t \right) +\tilde{V}\left( t \right) ,\tilde{\rho}\left( t \right) \right] \\ =-i\left[ \tilde{V}\left( t \right) ,\tilde{\rho}\left( t \right) \right] \]

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Further reading