The hydrogen ground state

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

Deriving the Spectrum, Ch Atoms in magnetic fields

Problem

The hydrogen 1s1s ground state, term symbol 2S1/2^2S_{1/2}, has nuclear spin I=12I=\tfrac12 (the proton) and hyperfine levels F=0,1F=0,1. Take gJ=2g_J=2 for this state. The proton's gg-factor is gI=5.5857g_I=5.5857 in nuclear magnetons, and μN/μB=me/mp=5.446×10−4\mu_{\rm N}/\mu_{\rm B}=m_e/m_p=5.446\times10^{-4}, so in Bohr magnetons the nuclear gg-factor is gI′=−gIme/mpg_I'=-g_Im_e/m_p, negative because the proton's moment points along its spin. A field of B=1.00 mTB=1.00\,\mathrm{mT} is applied, well below the 0.051 T0.051\,\mathrm T at which the Zeeman energy matches the hydrogen ground-state hyperfine interval. ((a)) Use the projection theorem, keeping the nuclear term, to find gFg_F for F=1F=1. ((b)) An oscillating magnetic field perpendicular to B⃗\vec B drives magnetic-dipole transitions between adjacent Zeeman sublevels of F=1F=1 (ΔMF=±1\Delta M_F=\pm1). At this field the quadratic part of the Breit-Rabi energies makes the two lines differ slightly in frequency. Find the mean of the two frequencies.

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