QS 7: Lippmann-Schwinger Equation

$\newcommand{\h}{{\mathcal H}}
\newcommand{\molp}{\Omega^{(+)}}
\newcommand{\molm}{\Omega^{(-)}}
\newcommand{\molpm}{\Omega^{(\pm)}}
\newcommand{\dydxt}[2]{\frac{d#1}{d#2}}
\newcommand{\dydx}[2]{\frac{\partial#1}{\partial#2}}$

We can easily prove
\begin{eqnarray}\molp=1+\frac{1}{i\hbar}\int_{-\infty}^0U_0(t)^{-1}V\molp U_0(t) dt\label{LS1}\end{eqnarray}
where $V\equiv H-H_0$.

Similarly, the S-matrix can be written as
\begin{eqnarray}S=1+\frac{1}{i\hbar}\int_{-\infty}^\infty U_0(t)^{-1}V\molp U_0(t) dt\end{eqnarray}
The first of these equations (a form of the Lippmann-Schwinger integral equation) is important because in most cases we have to depend on the perturbation theory for solving actual problems. This equation provides a way to calculate $\molp$ (and therefore also $S$ from the second equation) by iteration because $U_0(t)$, being the free evolution operator, is supposed to be known and $V$ can be considered as small in perturbation theory.

Proof:

Starting from the trivial identity
\begin{eqnarray*} \int_{-\infty}^0 \dydxt{}{t}[U(t)^{-1}U_0(t)] dt =1-\molp, \end{eqnarray*}
differentiate inside the integral and use $i\hbar dU/dt=HU$ as well as $i\hbar dU_0/dt=H_0U_0$. Therefore
\begin{eqnarray*} \molp=1+\frac{1}{i\hbar}\int_{-\infty}^0U(t)^{-1}VU_0(t) dt.\end{eqnarray*}
The adjoint of this equation gives
\begin{eqnarray*} {\molp}^\dagger=1-\frac{1}{i\hbar}\int_{-\infty}^0U_0(t)^{-1}VU(t) dt.\end{eqnarray*}
Multiply by $\molp$ on the right and use ${\molp}^\dagger\molp=1$ to get
\begin{eqnarray}\molp=1+\frac{1}{i\hbar}\int_{-\infty}^0U_0(t)^{-1}VU(t)\molp dt.\end{eqnarray}
Using (\ref{mol6}),
\begin{eqnarray}\molp=1+\frac{1}{i\hbar}\int_{-\infty}^0U_0(t)^{-1}V\molp U_0(t)dt.\end{eqnarray}

The proof of the formula for the S-matrix uses the following two steps. (Details are left as exercise.)

1. Write down the analog of the equation for $\molm$.
2. Show that
   \begin{eqnarray*} {\molp}^\dagger-{\molm}^\dagger=-\frac{1}{i\hbar}\int_{-\infty}^\infty U_0(t)^{-1}VU(t)dt, \end{eqnarray*}and hence\begin{eqnarray*} S=1+\frac{1}{i\hbar}\int_{-\infty}^\infty U_0(t)^{-1}V\molp U_0(t) dt\end{eqnarray*}

By operating on a free state $\phi(t)$ the Lippman-Schwinger equation(\ref{LS1}) can be written in the more familiar form,
\begin{eqnarray}\psi(t)=\phi(t)+\frac{1}{i\hbar}\int_{-\infty}^t
U_0(t-\tau)V\psi(\tau)d\tau \label{LS2}\end{eqnarray}
where $\psi(t)=\molp\phi(t)$ are the scattering states.
This equation is the starting point of evaluating the scattering states through iteration or other methods.