Question

Using the method of partial waves and phase shifts , prove the following general relation satisfied by the quantum scattering amplitude f(θ):

(47/k) * Im[f(0) = 0

where Im[· · ·] stands for the imaginary part and σ is the total quantum-mechanical scattering cross-section.

Answer 1

Quantum scattering amplitude f(theta) is given by = f(theta) = sigma^infinity_l = 0 f_l(theta) = 1/k sigma^inifity_l = 0 (2l + 1)e^is l sin S_l p_l(cos theta) Differential cross section is given by d sigma/d theta = 1f(theta)^2 = 1/k^2 sigma^inifinity_l = 0 sigma^infinity_l = 0 (2l + 1)(2l + 1)) e^i(s_l - S_l) sin L_l sin S_l p_l(cos theta)p_l(cos theta), sigma = > total cross section sigma = integral d sigma/dr dr = integral^pi_0 lf(theta))^2 sin theta d theta integral^2 pi_0 d phi \r\n 2pi integral^pi_0 lb(theta)^2 sin theta d theta = 2pi/k^2 sigma^inifinity_l = 0 sigma^inifinity_l = 0 (2l + 1)(2l'1 + 1)e^i(s_l - s_l') sins_l sins_l' integral^pi_0 p_l (cos theta)p_l'(cos theta) sin theta d theta integral^pi_0 p_l (cos theta)p_l' (cos theta) sin theta d theta = 2/2l + 1 S + l' sigma = sigma^inifinity + l = 0 sigma_l = 4 pi/k^2 sigma^inifinity_l = 0 (2l + 1) sin^c_l rightarrow (1) sigma_l are partial cross selection corresponding to scattering of particles in various angular momentum states. \r\n formulated scattering amplitude f(0) p_l (cos theta) = p_l(1) = 1 then