1 / 11

Energy Levels in Hydrogen

Energy Levels in Hydrogen. 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

stacey-reyes
Download Presentation

Energy Levels in Hydrogen

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Energy Levels in Hydrogen • 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 P460 - spin-orbit

  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): • convert back to lab frame (Thomas precession due to non-inertial frame gives a factor of 2). Energy depends on spin-orbit coupling P460 - spin-orbit

  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) P460 - spin-orbit

  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 - spin-orbit

  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 j=1/2 L=1 P460 - spin-orbit

  6. 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 - spin-orbit

  7. Relativistic Effects • Solved using non-relativistic S.E. can treat relativistic term (Krel) as a perturbation • <V> can use virial theorom P460 - spin-orbit

  8. 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) P460 - spin-orbit

  9. 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 E N=2 N=1 P460 - spin-orbit

  10. 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 - spin-orbit

  11. 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 - spin-orbit

More Related