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Problem 10.15 in the third edition of Griffith's Quantum Mechanics textbook asks to compute the scattering amplitude $f(\theta)$ for the low-energy scattering off the soft-sphere potential

$V(\vec r) = V_0 $ for $r \leq a$ and $0$ otherwise

in the second Born approximation. He gives the answer as

$-\left(2mV_0 a^3/3\hbar^2\right)\left[1-\left(4mV_0a^2/5\hbar^2\right)\right]$.

Fine, I did that and reproduced the result. However, from the optical theorem

$\int |f(\theta)|^2 d\Omega = \frac{4\pi}{k}\text{Im}[f(0)]$, I would have expected the amplitude at second order to have an imaginary part (since the leading-order cross section is second order in the coupling $V_0$).

How then is this answer compatible with the optical theorem? I would not expect that taking the low-energy limit would make us lose this important information. If that is the case, how so?

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  • $\begingroup$ FYI to answerers: A hard sphere mean $V_0=\infty$, it does not mean a smooth transition to $V=0$ around $r=a$. $\endgroup$
    – JEB
    Jul 13, 2019 at 23:41

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