Regarding the Born approximation

[Kontilera]

Hello!
In order to prepare for an exam I have started solving exercies problems and have gotten most of them right but have quetion a regarding a solution.

In this probelm I used the first order Born approximation in order to calculate the differential and total cross section for the potential,
V(r) = \lambda e^{-r^2 / 4a^2}.
Now, how can I see the limits of this approximation? E.g. can we see if it is valid for a particle with a given mass assuming that we have ka \ll 1?

 

[DrClaude]

The criterion for the validity of the Born approximation is
$$
\frac{m | V_0 | a^2}{\hbar^2} \ll 1
$$
where $m$ is the mass of the particle, $V_0$ and $a$ the height and range of the potential, respectively.