From Yukawa Potential to Rutherford Scattering

The starting point it represents the Klein-Gordon equation of Section 2.2.3, Eq. (2.34) in the situation the wave-function solution is represented by the potential (its source) in a spherical (as a source) symmetry. 

Thorough transforming the Cartesian Laplacian into the radial one through the recipe of the Section 3.3.1, 

considering the other extreme limit, namely the behavior in the origin, one observes that for r 0 the term with 2 r dominates the entire equation, thus suggesting the specific dependence  

Quantum Nanochemistty-Volume I Quantum Theory and Observability 

Overall, combining the two potential forms in a continuous analytical formulation one establishes the so-called Yukawa potential 

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