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Modern physics and Quantum Mechanics Physical Systems, 8 Mar.2007 EJZ

Modern physics and Quantum Mechanics Physical Systems, 8 Mar.2007 EJZ. More angular momentum and H atom Compare to Bohr atom Applications: Bohr magneton, Zeeman effect Brief review of modern physics and QM Conferences next week Next quarter. Quantization of angular momentum.

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Modern physics and Quantum Mechanics Physical Systems, 8 Mar.2007 EJZ

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  1. Modern physics and Quantum MechanicsPhysical Systems, 8 Mar.2007 EJZ • More angular momentum and H atom • Compare to Bohr atom • Applications: Bohr magneton, Zeeman effect • Brief review of modern physics and QM • Conferences next week • Next quarter

  2. Quantization of angular momentum Show that for ANY radial potential V(r) in the spherical Schrödinger equation, both the total angular momentum and the z-component are quantized. Last week we discussed the momentum operators…

  3. Spherical harmonics solve spherical Schrödinger equation for any V(r)

  4. Possible orientations of L and Lz (for l=2) Example 7.1 (p.300), #7.12, 7.14 (p.332)

  5. H-atom: quantization of energy for V= - kZe2/r Solve the radial part of the spherical Schrödinger equation (next quarter): Do these energy values look familiar?

  6. QM H-atom energy levels: degeneracy for states with different qn and same energy Selections rules for allowed transitions: l must change by one, since energy hops are mediated by a photon of spin-one. Dn = anything Dm can = ±1 or 0

  7. H-atom: wavefunctionsY(r,q,f) for V= - kZe2/r We already have the angular part of the wavefunctions for any radial potential in the spherical Schrödinger equation: We can solve (next quarter) for R(r) ~ Laguerre Polynomials

  8. H-atom wavefunctions ↔ electron probability distributions Discussion: compare Bohr model to Schrödinger model for H atom.

  9. A fourth quantum number: intrinsic spin If there are 2s+1 possible values of ms, and only 2 orientations of ms = z-component of s (Pauli), What values can s and ms have?

  10. Stern-Gerlach showed splitting due to spin, even when l=0 l = 1, m = 0, ±1 l = 0, m = ±1/2

  11. Spinning particles shift energies in B fields Cyclotron frequency: An electron moving with speed v perpendicular to an external magnetic field feels a Lorentz force: F=ma (solve for w=v/r) Solve for Bohr magneton…

  12. Magnetic moments shift energies in B fields

  13. Spin S and orbit L couple to total angular momentum J = L + S

  14. Spin-orbit coupling: spin of e- in magnetic field of pFine-structure splitting (e.g. 21-cm line) (Interaction of nuclear spin with electron spin (in an atom) → Hyper-fine splitting)

  15. Total J + external magnetic field → Zeeman effect

  16. Total J + external magnetic field → Zeeman effect

  17. Total J + external magnetic field → Zeeman effect

  18. History of Light quantization • Stefan-Boltzmann blackbody had UV catastrophe • Planck quantized light, and solved blackbody problem • Einstein used Planck’s quanta to explain photoelectric effect • Compton effect demonstrated quantization of light • Corrollary: deBroglie’s matter waves, discovered by Davisson & Germer hc/l = Kmax + F

  19. History of atomic models: • Thomson discovered electron, invented plum-pudding model • Rutherford observed nuclear scattering, invented orbital atom • Bohr quantized angular momentum, for better H atom model. • Bohr model explained observed H spectra, derived En = E/n2 and phenomenological Rydberg constant • Quantum numbers n, l, ml (Zeeman effect) • Solution to Schrodinger equation showed that En = E/l(l+1) • Pauli proposed spin (ms=1/2), and Dirac derived it

  20. deBroglie’s matter waves  Bohr’s angular momentum quantization Compton Effect

  21. Quantum wells

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