Principles of Quantum Mechanics · Problem 19.6

Scattering Theory — R. Shankar

Physics — Quantum Mechanics · partial-wave-analysis

Problem

Prove the optical theorem:

\sigma_{\text{tot}} = \frac{4\pi}{k} \text{Im}\, f(0)

starting from the partial wave expansion

f(\theta) = \frac{1}{k} \sum_{l=0}^{\infty} (2l+1) e^{i\delta_l} \sin\delta_l \, P_l(\cos\theta)

and using the orthogonality of Legendre polynomials.