Principles of Quantum Mechanics · Problem 19.2

The optical theorem connects the imaginary part of the forward scattering amplitude to the total cross section—a deep result linking destructive interference (forward scattering) to overall absorption of the incident wave. For a hard sphere in the Born approximation, you'll compute the scattering amplitude $f(\theta)$ explicitly, extract its imaginary part at $\theta=0$, then verify it matches the integrated differential cross section $\int |f(\theta)|^2 d\Omega$. This check reveals how quantum unitarity constrains scattering: the forward amplitude's phase shift accounts for all particles removed from the beam.

The optical theorem connects the imaginary part of the forward scattering amplitude to the total cross section—a deep result linking destructive interference (forward scattering) to overall absorption of the incident wave. For a hard sphere in the Born approximation, you'll compute the scattering amplitude $f(\theta)$ explicitly, extract its imaginary part at $\theta=0$, then verify it matches the integrated differential cross section $\int |f(\theta)|^2 d\Omega$. This check reveals how quantum unitarity constrains scattering: the forward amplitude's phase shift accounts for all particles removed from the beam.