Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / Section A / 5 / Solution

Write $K\propto m^{\alpha }{R}^{\beta }{h}^{\delta }$. Matching dimensions to $M{L}^2{T}^{-2}$ givesThe dimensionless factor $C$ cancels when the same confinement geometry and state are scaled. HenceUsing $m_p/m_e\simeq 1836$ gives this value. It remains small compared with the proton rest energy, so the nonrelativistic mass-radius scaling is self-consistent.