Classical Mechanics (Goldstein) · Problem 32

The Central Force Problem — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · rectangular-well-scattering-refraction

Problem

A central force potential frequently encountered in nuclear physics is the rectangular well, defined by the potential V=0r>aV = 0 \quad r > a =−V0r≤a.= -V_0 \quad r \leq a. Show that the scattering produced by such a potential in classical mechanics is identical with the refraction of light rays by a sphere of radius aa and relative index of refraction n=E+V0E.n = \sqrt{\frac{E + V_0}{E}}. (This equivalence demonstrates why it was possible to explain refraction phenomena both by Huygen's waves and by Newton's mechanical corpuscles.) Show also that the differential cross section is σ(Θ)=n2a24cos⁡Θ2(ncos⁡Θ2−1)(n−cos⁡Θ2)(1+n2−2ncos⁡Θ2)2.\sigma(\Theta) = \frac{n^2 a^2}{4\cos\frac{\Theta}{2}} \frac{\left(n\cos\frac{\Theta}{2} - 1\right)\left(n - \cos\frac{\Theta}{2}\right)}{\left(1 + n^2 - 2n\cos\frac{\Theta}{2}\right)^2}. What is the total cross section?

Related theory

Browse the problem library