For the hydrogenic level ( , , )

PHYSICS · Subjective · 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

For the hydrogenic level 4f5/24f_{5/2} (l=3l=3, s=12s=\tfrac12, j=52j=\tfrac52), use the projection theorem for a vector operator within a level, j(j+1)ℏ2⟨V⃗⟩=⟨V⃗⋅J⃗ J⃗⟩,j(j+1)\hbar^2\langle\vec V\rangle = \langle\vec V\cdot\vec J\,\vec J\rangle, applied to V⃗=S⃗\vec V = \vec S, together with S⃗⋅J⃗=12(J2−L2+S2)\vec S\cdot\vec J = \tfrac12(J^2 - L^2 + S^2), to calculate ⟨Sz⟩\langle S_z\rangle in the state mj=5/2m_j = 5/2. Give the answer as a multiple of ℏ\hbar.

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