[NOTES/QM-18005]Born Approximation

Contents

1. Perturbative solution of integral equation
2. First Born Approximation 

Perturbative solution of integral equation

 The energy eigen functions, with a correct asymptotic behaviour corresponding to the scattering solutions satisfy the following integral equation \begin{equation} \psi(\vec{r})=\exp(i\vec{k_i}.\vec{r})-\frac{1}{4\pi} \int\frac{\exp\{ik|\vec{r}-\vec{r}^{\,{'}}|\}}{|\vec{r}-\vec{r}^{\,{'}}|} U(\vec{r}^{\,{'}})\psi (\vec{r}^{\,{'}}) d^3 r{'} . \label{E1} \end{equation} where we have used the notation

• $\vec{k}_i $ = momentum of the incident beam
•  $\vec{k}_f$ = momentum of the scattered beam
•  $\theta$= scattering angle = angle bewteen $\vec{k}_i$ and $\vec{k}_f.$
•  $V(\vec{r}) $ = potential due the target
•  $\mu$= mass of the incident particle ( reduced mass for two body problem)
•  $U(\vec{r})$= $ \dfrac{2\mu}{\hbar^2}V(\vec{r}). $

An iterative solution of the integral equation can be obtained by assuming that in the lowest order approximation $\psi(\vec{r})$ is equal to $\psi_0(\vec{r})$ given by \begin{equation} \psi_0(\vec{r})= \exp(i\vec{k}_i\cdot\vec{r}) \label{E2} \end{equation} Using this approximation for $\psi(\vec{r})$ from \eqref{E2} in the right hand side of \eqref{E1}, we get the next order solution, denoted as $\psi_1(\vec{r})$, given by \begin{equation} \psi_1(\vec{r}) = \exp(i\vec{k}_i\cdot\vec{r}) - \frac{1}{4\pi} \int\frac{\exp\{ik|\vec{r}-\vec{r}^{'}|\}}{|\vec{r}-\vec{r}^{'}|} U(\vec{r}^{'})\exp(i\vec{k}_i\cdot\vec{r}^{'}) d^3 r{'} .\label{E3} \end{equation} The next approximation to the solution, $\psi_2(\vec{r})$, is obtained by replacing $\psi(\vec{r})$ in \eqref{E1} with $\psi_1(\vec{r})$. 

Thus
\begin{eqnarray}\psi_2(\vec{r}) &=&e^{i\vec{k_i}.\vec{r}}-\frac{1}{4\pi} \int\frac{e^{ik|\vec{r}-\vec{r}^{'}|}}{|\vec{r}-\vec{r}^{'}|} U(\vec{r}^{'})\psi_1(\vec{r}^{'}) d^3r{'} \\&=& e^{i\vec{k}_i\cdot\vec{r}} - \frac{1}{4\pi} \int\frac{e^{ik|\vec{r}-\vec{r}^{'}|}}{|\vec{r}-\vec{r}^{'}|} U(\vec{r}^{'})e^{(i\vec{k}_i\cdot\vec{r}^{'})} d^3 r{'}\\ && + \frac{1}{(4\pi)^2}\int \frac{e^{ik|\vec{r}-\vec{r}^{'}|}}{|\vec{r}-\vec{r}^{'}|} U(\vec{r^{'}})\int \exp(i\vec{k}_i\cdot\vec{r^{'}}) \frac{e^{ik|\vec{r}^{'}-\vec{r}^{{''}}|}}{|\vec{r^{'}}-\vec{r^{{''}} }|} U(\vec{r}^{''})e^{i\vec{k}_i\cdot(\vec{r}^{'}+\vec{r}^{''})}d^3r^{'} d^3r^{''}. \label{E4} \end{eqnarray}

 This process can be continued indefinitely and it becomes very cumbersome to compute the wave function beyond first few orders

First Born Approximation 

The first order Born approximation consists in using the first, the plane wave term in the above series as approximate wave function in the expression \begin{eqnarray} f(\theta,\phi) &=& -\frac{1}{4\pi} \int \exp(-i\vec{k}_f\cdot\vec{r}^{'})U(r^{'}) \psi(r^{'}) d^3r^{'} \label{E20} \\ &=& -\left(\frac{\mu}{2\pi \hbar^2}\right)\int \exp(-i\vec{k}_f\cdot\vec{r}^{'})V(r^{'}) \psi(r^{'}) d^3r^{'} \label{E5} \end{eqnarray} for the scattering amplitude, giving \begin{equation} f(\theta,\phi)\approx -\frac{\mu}{2\pi\hbar^2}\int e^{i(\vec{k}_i-\vec{k}_f) \cdot\vec{r}} V(r) d^3 r \label{E6}, \end{equation} or \begin{equation} f(\theta,\phi)\approx -\frac{\mu}{2\pi\hbar^2}\int e^{-i\vec{q} \cdot\vec{r}} V(r) d^3r \label{E7}, \end{equation} where $\vec{q}=\vec{k}_i-\vec{k}_f$ is the momentum transfer. The result \eqref{E7} is the well known, first order, Born approximation result for the scattering amplitude.