# 12.8: General Notes on Scattering in the Born Approximation


The square well illustrates some general feature of scattering in the Born approximation:

• Born approximation is based on perturbation theory, so works best for high energy particles.
• Scattering depends on $$V^2_0$$, so both attractive and repulsive potentials behave the same.
• At high energy, cross section is inversely proportional to the energy $$(E = \hbar^2 k^2/2m)$$
• Dependence on $$k$$ and $$\theta$$ arises only through the combination $$\chi = 2k \sin \frac{\theta}{2}$$. Thus as energy increases, the scattering angle $$\theta$$ is reduced and the scattered beam becomes more peaked in the ‘straight on’ direction.
• Angular dependence depends on the range of the potential $$a$$ but not on the strength $$V_0$$.
• Total cross section depends on both range $$a$$ and depth $$V_0$$ of the potential.

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