Treat He ( ) as a hydrogenic ion with its reduced mass taken equal to

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

Deriving the Spectrum, Ch The exact Coulomb problem and Rydberg states

Problem

Treat He+^+ (Z=2Z=2) as a hydrogenic ion with its reduced mass taken equal to mem_e, so that En=−Z2(13.6057 eV)/n2E_n=-Z^2(13.6057\,\mathrm{eV})/n^2. Take α=1/137.036\alpha=1/137.036 and e2/(4πϵ0a0)=27.2114 eVe^2/(4\pi\epsilon_0a_0)=27.2114\,\mathrm{eV}. ((a)) For n=3n=3, use the virial theorem to split E3E_3 into ⟨T^⟩\langle\hat T\rangle and ⟨V⟩\langle V\rangle, in eV. ((b)) Check ⟨V⟩\langle V\rangle independently from the radial moment ⟨r−1⟩\langle r^{-1}\rangle. ((c)) Find the root-mean-square speed of the electron as a fraction of cc, and compare it with the speed in hydrogen's n=3n=3 level.

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