[NOTES/QM-18007] Validity of Born Approximation Square Well Potential

For a square well potential of strength $V_0$ and range $R_0$, the expression in the left hand side \eqref{E9} takes the form
\begin{eqnarray} \frac{\mu V_0}{\hbar^2 k}\left|\int_0^{R_0} \left( e^{2ikr}-1\right) dr\right|  &=& \frac{\mu V_0}{\hbar^2 k}\left|\frac{e^{2ikR_0}-1}{2ik} - R_0 \right|\label{E2}\\ &=& \frac{\mu V_0}{2\hbar^2 k^2}\left|e^{2ikR_0}- 2ik R_0 -1 \right|\label{E3} \end{eqnarray}Using the notation $\rho=2kR_0$ the condition, that the Born approximation be valid, takes the form \begin{equation} \frac{\mu V_0}{2\hbar^2k^2}\left( \rho^2 -2\rho\sin\rho -2\cos\rho +2 \right)^{\frac{1}{2}} << \label{E4} \end{equation} we shall consider the low energy and high energy cases separately.

Low energy scattering

At low energy the de Broglie wave length is much larger than the range of the potential {\it i.e.} $2kR_0 << 1$. We then have \begin{eqnarray} \rho^2 &-&2\rho\sin\rho + 2\cos\rho \nonumber\\ &\approx& \rho^2 -2\rho\left(\rho-\frac{\rho^3}{6} +\cdots)\right) - 2 \left( 1-\frac{\rho^2}{2} +\frac{\rho^4}{24} + \cdots \right) +2 \nonumber \\ &= &\frac{\rho^4}{4}\label{E5} \end{eqnarray} Hence at low energies the Born approximation is applicable if \begin{equation} \frac{\mu V_0}{2\hbar^2k^2}\frac{\rho^2}{2}= \frac{\mu V_0 R_0^2}{\hbar^2} << 1 \label{E6} \end{equation} The above condition implies that the potential is so weak that the bound state does not exist.

High energy limit

In the high energy limit $\rho >> 1$ and we get \begin{equation} \left( \rho^2 -2\rho\sin\rho -2\cos\rho +2 \right)^{\frac{1}{2}} \approx \rho\label{E7} \end{equation} and hence the Born approximation is valid if \begin{equation} \frac{\mu V_0}{2\hbar^2k^2} \rho <<1 \label{E8} \end{equation} or \begin{equation} \frac{\mu V_0 a}{\hbar v} << 1 \label{E9} \end{equation}