Let q = k₀ − k be the scattering vector defined in Example 1.2. If a_B is the Bohr radius, show that the cross section for plane wave scattering from a hydrogen atom is proportional to the factor [1 + (qa_B/2)²]⁻⁴.

Example 1.2:

Prove that ∇ × (A × B) = A∇ · B − (A · ∇)B + (B · ∇)A − B∇ · A.

Solution:

Focus on the ith Cartesian component and use the left side of (1.37) to write

The cyclic properties of the Levi-Civita symbol and the identity (1.39) give 

Therefore,

This proves the identity because the choice of i is arbitrary.