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When studying scattering in quantum mechanics, we define the "in" and "out" states as states that are eigenfunctions of the free Hamiltonian at ${t \rightarrow \mp\infty}$. This makes sense to me Schrodinger picture where we can start with free states at ${t \rightarrow -\infty}$ which then evolve with time under the total Hamiltonian (with the interaction) and then go back to free states at ${t \rightarrow +\infty}$.

What I don't understand is how to define these "in" and "out" states in the Heisenberg picture. I have been trying to read about this from Weinberg's book on quantum mechanics but it is not very clear to me.

Suppose we define it in the same way as the Schrodinger picture, then the states don't time evolve here but neither does the Hamiltonian. So it seems that we need to introduce another operator that time evolves to define these states properly in the Heisenberg picture.

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The Hamiltonian does change, as the free Hamiltonian that defines the states at $t\to \pm\infty$ is not the interacting Hamiltonian that acts about $t=0$. The assumption is that the states evolve as we turn on (and then off) the interaction adiabitically $$ \frac{d}{dt}\mathcal{H} = \frac{\partial H}{\partial t} \propto V $$

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