A hypothetical excited atomic level has total angular momentum

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Radiative transitions, external fields and collisions

Deriving the Spectrum, Ch Atoms in electric fields

Problem

A hypothetical excited atomic level has total angular momentum J=2J=2. Second-order perturbation theory in an applied field gives it a scalar polarisability α0=−2.0×10−40 C m2 V−1\alpha_0=-2.0\times10^{-40}\,\mathrm{C\,m^2\,V^{-1}} and a tensor polarisability α2=+4.0×10−40 C m2 V−1\alpha_2=+4.0\times10^{-40}\,\mathrm{C\,m^2\,V^{-1}} (the negative scalar value reflects that this excited level, unlike a ground state, is on average pushed upward by nearby states below it). A static field E=1.0×106 V m−1\mathcal E = 1.0\times10^6\,\mathrm{V\,m^{-1}} is applied along z^\hat z, and the sublevels shift according to ΔEJM=−12E2[α0+α2 3M2−J(J+1)J(2J−1)].\Delta E_{JM} = -\tfrac12\mathcal E^2\Big[\alpha_0+\alpha_2\,\frac{3M^2-J(J+1)}{J(2J-1)}\Big]. ((a)) Before calculating anything, state which pairs of sublevels must remain exactly degenerate, and why. ((b)) Calculate ΔE/h\Delta E/h, in MHz, for each distinct sublevel M=0,±1,±2M=0,\pm1,\pm2. ((c)) State whether every sublevel moves in the same direction, and contrast this with the rule that a ground state's Stark shift is always downward.

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