IPhO_2025_Q1-04

Hydrogen orbital energy (A.3)

IPhO · 2025 · Fisika Modern

Hydrogen and galaxies (Part A — Introduction; official 2025 IPhO question). This problem studies hydrogen and the mass distribution of galaxies. In the Bohr model, a non-relativistic electron of mass m_{e} orbits a fixed proton on a circular path. Throughout this part, use ℏ = h / (2 π) . The electron is non-relativistic and orbits a fixed proton on a circular orbit. The Bohr model quantizes angular momentum as L = n ℏ, where n is a positive integer; α = e^{2} / (4 π ϵ_{0} ℏ c) \approx_{7} .27 \times 1 0^{- 3} . Determine the electron's mechanical energy E_{n} on an orbit of radius r_{n} .

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