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Part of JPH-02 — Atoms: Bohr Model & Hydrogen Spectrum

Reduced Mass Correction

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−∗∗Tags∗∗:reduced−mass,correction,isotope−∗∗Difficulty∗∗:AdvancedIntheBohrmodel,theelectronmassmshouldbereplacedbythereducedmassμ=mM/(m+M),whereMisthenuclearmass.Forhydrogen:μ=m×1836m/(m+1836m)=m×1836/1837≈0.99946m.Thecorrectionistiny( 0.05- **Tags**: reduced-mass, correction, isotope - **Difficulty**: Advanced In the Bohr model, the electron mass m should be replaced by the reduced mass μ = mM/(m+M), where M is the nuclear mass. For hydrogen: μ = m × 1836m/(m + 1836m) = m × 1836/1837 ≈ 0.99946m. The correction is tiny (~0.05%) but detectable. For positronium ($e^{+}$$e^{-}$): μ = m/2, so all energies are halved and radii are doubled compared to hydrogen. For muonic hydrogen (muon replaces electron, m_μ = 207m_e): μ ≈ 186m_e, so the radii are ~200× smaller and energies ~200× larger. JEE occasionally tests the reduced mass concept for exotic atoms.$

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