A hydrogenic ion with a single electron and nuclear charge (point nucleus

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

A hydrogenic ion with a single electron and nuclear charge Z=80Z=80 (point nucleus, infinite nuclear mass) has 1s1s binding energy given exactly, for the ground state, by the Dirac formula EbindDirac=mc2(1−1−(Zα)2),E_{\rm bind}^{\rm Dirac} = mc^2\left(1-\sqrt{1-(Z\alpha)^2}\right), with mc2=511 keVmc^2 = 511\,\mathrm{keV} and α=1/137.036\alpha=1/137.036. ((a)) Calculate EbindDiracE_{\rm bind}^{\rm Dirac}. ((b)) Calculate the binding energy predicted by the fine-structure formula, correct to relative order (Zα)2(Z\alpha)^2, E1,1/2=E1[1+(Zα)2(11/2+1/2−34)]E_{1,1/2} = E_1\left[1+(Z\alpha)^2\left(\dfrac{1}{1/2+1/2}-\dfrac34\right)\right], with E1=−13.6 Z2 eVE_1 = -13.6\,Z^2\,\mathrm{eV}. ((c)) State, from the size of the discrepancy, whether the perturbative (Zα)2(Z\alpha)^2 treatment is still trustworthy at this ZZ.

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