 Solution

ID: 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^{α} R^{β} h^{δ}

. Matching dimensions to

M L^{2} T^{- 2}

gives

δ = 2, α = - 1, β = - 2, K = C \frac{h ^{2}}{m R ^{2}} .

The dimensionless factor

C

cancels when the same confinement geometry and state are scaled. Hence

K_{p} = K_{e} \frac{m _{e}}{m _{p}} (\frac{R _{e}}{R _{p}})^{2}, K_{p} ≃ 8.3 MeV .

Using

m_{p} / m_{e} ≃ 1836

gives this value. It remains small compared with the proton rest energy, so the nonrelativistic mass-radius scaling is self-consistent.

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