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QS 3: States that are free in the past

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$\newcommand{\h}{{\mathcal H}}$

We now come to the central concept of the scattering theory. What does it mean when we say that a particle moving under the influence of a potential $V$ looks like a free particle in the remote past?

Let us consider two systems. One, the actual system, with hamiltonian $H$. Its states are represented by vectors $\psi$ etc. in a Hilbert space $\h$. The other is a fictitious,  free system, with hamiltonian $H_0$, and states represented by vectors in the same common Hilbert space.

Let $\psi$ be the state of the particle at some given time, say  $t=0$. Then the state at time $t$ is
\begin{eqnarray*} \psi(t)=U(t)\psi\qquad U(t)=\exp(-iHt/\hbar) \end{eqnarray*}
A vector  $\phi \in \h$ would represent the sate of a free particle at $t=0$ if, at any other time $t$, it were given by
\begin{eqnarray*} \phi(t)=U_0(t)\psi\qquad U_0(t)=\exp(-iH_0t/\hbar). \end{eqnarray*}


Now suppose that $\psi(t)$ is such that it becomes {\em indistinguishable} from $\phi(t)$ (for some $\phi$) for large negative $t$ then we can say that the particle indeed behaves like a free particle in remote past, that is, if
\begin{eqnarray*} \lim_{t\to -\infty}\psi(t)=\lim_{t\to -\infty}\phi(t). \end{eqnarray*}

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