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The Real Hydrogen Atom

The Real Hydrogen Atom. Solve SE and in first order get (independent of L): can use perturbation theory to determine: magnetic effects (spin-orbit and hyperfine e-A) relativistic corrections

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The Real Hydrogen Atom

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  1. The Real Hydrogen Atom • Solve SE and in first order get (independent of L): • can use perturbation theory to determine: • magnetic effects (spin-orbit and hyperfine e-A) • relativistic corrections • Also have Lamb shift due to electron “self-interaction”. Need QED (Dirac eq.) and depends on H wavefunction at r=0 (source of electric field). Very small and skip in this course (first calculated by Bethe using perturbation theory on train from Long Island to Ithaca. Bethe also in film Fat Man and Little Boy....) P460 - real H atom

  2. Spin-Orbit Interactions • A non-zero orbital angular momentum L produces a magnetic field • electron sees it. Its magnetic moment interacts giving energy shift • in rest frame of electron, B field is (see book/ER): • convert back to lab frame (Thomas precession due to non-inertial frame gives a factor of 2 – Dirac eq gives directly). Energy depends on spin-orbit coupling P460 - real H atom

  3. Spin-Orbit: Quantum Numbers • The spin-orbit coupling (L*S) causes ml and ms to no longer be “good” quantum numbers • spin-orbit interactions changes energy. • In atomic physics, small perturbation, and can still use H spatial and spin wave function as very good starting point. Large effects in nuclear physics (and will see energy ordering very different due to couplings). So in atomic just need expectation value of additional interaction P460 - real H atom

  4. SL Expectation value • Determine expectation value of the spin-orbit interaction using perturbation theory. Assume J,L,S are all “good” quantum numbers (which isn’t true) • assume H wave function is ~eigenfunction of perturbed potential P460 - real H atom

  5. SL Expectation value • To determine the energy shift, also need the expectation value of the radial terms using Laguerre polynomials • put all the terms together to get spin-orbit energy shift. =0 if l=0 J=3/2, L=1 J=1/2 L=0 J=1/2 L=1 with relativistic n=2 L=0,1 j=3/2 j=1/2 P460 - real H atom

  6. Numerology • have • but • and so P460 - real H atom

  7. Spin Orbit energy shift • For 2P state. N=2, L=1, J= 3/2 or 1/2 • and so energy split between 2 levels is J=3/2 j=1/2 L=1 P460 - real H atom

  8. Relativistic Effects • Solved using non-relativistic S.E. can treat relativistic term (Krel) as a perturbation • <V> can use virial theorem P460 - real H atom

  9. Relativistic+spin-orbit Effects • by integrating over the radial wave function • combine spin-orbit and relativistic corrections • energy levels depend on only n+j (!). Dirac equation gives directly (not as perturbation). For n=2 have: P460 - real H atom

  10. Energy Levels in Hydrogen # states N=3 • Degeneracy = 2j+1 • spectroscopic notation: nLj with L=0 S=state, L=1 P-state, L=2 D-state • also can note spin “doublet” is single electron with s=1/2 E N=2 N=1 P460 - real H atom

  11. Zeeman Effect:External B Field • Energy shift depends on mj and removes any remaining degeneracy. Now two fields (internal and external) and details of splitting depends on relative strengths • Unless S=0, the magnetic moment and the total angular momentum are not in the same direction (and aren’t in B direction). For weak external field, manipulating the dot products gives B>0 B=0 P460 - real H atom

  12. Zeeman Effect strong field DL=+-1 Dm=0,+1,-1 P460 - real H atom

  13. Zeeman Effect:External B Field • Assume that weak B field (if strong then L and S won’t couple) • B field off  1 photon energy B field on  6 photon energies (with their energy depending on the g factor and on the B field • One of the first indicators that the electron had intrinsic angular momentum s=1/2 B>0 B=0 P460 - real H atom

  14. Hyperfine Splitting • Many nuclei also have spin • p,n have S=1/2. Made from 3 S=1/2 quarks (plus additional quarks and antiquarks and gluons). G-factors are 5.58 and -3.8 from this (-2 for electron). • Nuclear g-factors/magnetic moments complicated. Usually just use experimental number • for Hydrogen. Let I be the nuclear spin (1/2) • have added terms to energy. For S-states, L=0 and can ignore that term P460 - real H atom

  15. Hyperfine Splitting • Electron spin couples to nuclear spin • so energy difference between spins opposite and aligned. Gives 21 cm line for hydrogen (and is basis of NMR/MRI) P460 - real H atom

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