# Total hamiltonian is time independent in interaction picture

There is this general statement in Ashok which, if it's true, could someone explain why it is true?

Regarding the interaction picture:

Since $$H_0$$ is time independent in the interaction picture (see (6.51)), it follows that the total Hamiltonian is time independent in the interaction picture.

I don't think this is right, but I can be wrong. The point is that in the Schrödinger picture

$$H^{(S)} = H_0^{(S)} + H_I^{(S)},$$

and then, in the interaction picture

$$$$\begin{split} H^{(IP)} &= e^{iH_0^{(IP)}t}(H_0^{(S)} + H_I^{(S)})e^{-iH_0^{(IP)}t} \\ &= H_0^{(IP)} + e^{iH_0^{(IP)}t}H_I^{(S)}e^{-iH_0^{(IP)}t}. \end{split}$$$$

Obviously, $$H_0$$ is time independent but in general, the interaction part $$H_I$$ is not, so I don't see why $$H$$ should be time-indepdendent as claimed.