The relativistic and spin-orbit averages for a hydrogenic level are H_ rel =- E_n^2…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure

Deriving the Spectrum, Ch Fine structure and the Lamb shift

Problem

The relativistic and spin-orbit averages for a hydrogenic level are ⟨Hrel⟩=−En22mc2[4nl+12−3]\langle H_{\rm rel}\rangle=-\frac{E_n^2}{2mc^2}\left[\frac{4n}{l+\frac12}-3\right] and ⟨Hso⟩=En2mc2 n[j(j+1)−l(l+1)−34]l(l+12)(l+1)(l≥1).\langle H_{\rm so}\rangle=\frac{E_n^2}{mc^2}\,\frac{n\left[j(j+1)-l(l+1)-\frac34\right]}{l(l+\frac12)(l+1)}\qquad(l\ge1). ((a)) For j=l−12j=l-\tfrac12, add the two and show that the sum is En22mc2(3−4nj+12)\dfrac{E_n^2}{2mc^2}\left(3-\dfrac{4n}{j+\frac12}\right), with no ll left. ((b)) For hydrogen, with En=−13.6057 eV/n2E_n=-13.6057\,\mathrm{eV}/n^2 and mc2=511.00 keVmc^2=511.00\,\mathrm{keV}, evaluate ⟨Hrel⟩\langle H_{\rm rel}\rangle, ⟨Hso⟩\langle H_{\rm so}\rangle and their sum for 3D3/23D_{3/2}, and the same three numbers for 3P3/23P_{3/2}, in μeV\mu\mathrm{eV}.

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