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Physics 451

Physics 451. Quantum mechanics I Fall 2012. Dec 5, 2012 Karine Chesnel. Homework. Final exam. Wednesday Dec 12 , 2012 7am – 10am C 285. Quantum mechanics. Last assignment HW 24 Thursday Dec 6 5.15, 5.16, 5.18, 5.19. 5.21. Quantum mechanics. Class evaluation. Please fill

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Physics 451

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  1. Physics 451 Quantum mechanics I Fall 2012 Dec 5, 2012 Karine Chesnel

  2. Homework Final exam Wednesday Dec 12, 2012 7am – 10am C 285 Quantum mechanics • Last assignment • HW 24 Thursday Dec 6 • 5.15, 5.16, 5.18, 5.19. 5.21

  3. Quantum mechanics Class evaluation Please fill the class evaluation survey online Quiz 34: 5 points

  4. Solids e- Pb 5.15: Relation between Etot and EF Pb 5.16: Case of Cu: calculate EF , vF, TF, and PF Fermi surface Bravais k-space Quantum mechanics

  5. Free electron gas Fermi surface Total energy contained inside the Fermi surface Quantum mechanics Bravais k-space

  6. Free electron gas Fermi surface Quantum mechanics Solid Quantum pressure Bravais k-space

  7. Solids e- Fermi surface Bravais k-space Number of unit cells Quantum mechanics

  8. Solids Bloch’s theorem Quantum mechanics Dirac comb V(x)

  9. Solids Quantum mechanics Circular periodic condition V(x) x-axis “wrapped around”

  10. Solids Quantum mechanics Solving Schrödinger equation V(x) a 0

  11. Solids or Quantum mechanics Boundary conditions V(x) a 0

  12. Solids • Discontinuity of Quantum mechanics Boundary conditions at x = 0 V(x) a 0 • Continuity of Y

  13. Solids Band structure Quantum mechanics Quantization of k: Pb 5.18 Pb 5.19 Pb 5.21

  14. Quiz 33 Quantum mechanics In the 1D Dirac comb model how many electrons can be contained in each band? A. 1 B. 2 C. q D. Nq E. 2N

  15. Solids Insulator: band entirely filled ( even integer) 2N electrons (2e in each state) Quantum mechanics Quantization of k: Band structure E Conductor: band partially filled N states Band Gap Semi-conductor: doped insulator N states Band Gap N states Band

  16. Quiz 33 Quantum mechanics A material has q=3 valence electrons / atoms. In which category will it fall according to the 1D dirac periodic potential model? A. Conductor B. Insulator C. Semi-conductor

  17. Quantum mechanics Final Review What to remember?

  18. “Operator” x “Operator” p Quantum mechanics Wave function and expectation values

  19. Stationary state General state Quantum mechanics Time-independent Schrödinger equation Here The potential is independent of time

  20. Excited states Quantization of the energy Ground state Quantum mechanics Review I Infinite square well 0 a x

  21. V(x) • Operator position • Operator momentum x Quantum mechanics Harmonic oscillator

  22. Raising operator: Lowering operator: Quantum mechanics Review I 4. Harmonic oscillator Ladder operators:

  23. Quiz 35 Quantum mechanics What is the result of the operation ? A. B. C. D. E. 0

  24. Physical considerations Symmetry considerations Quantum mechanics Square wells and delta potentials V(x) Scattering States E > 0 x Bound states E < 0

  25. is continuous is continuous is continuous is continuous except where V is infinite y a æ ö d 2 m ( ) D = - y ç ÷ 0 h 2 dx è ø Quantum mechanics Ch 2.6 Square wells and delta potentials Continuity at boundaries Delta functions Square well, steps, cliffs…

  26. Scattering state A F B x 0 Transmission coefficient Reflection coefficient “Tunneling” Quantum mechanics The delta function well/ barrier

  27. Wave function Vector Operators Linear transformation (matrix) is an eigenvector of Q Observables are Hermitian operators Quantum mechanics Formalism l is an eigenvalueof Q

  28. Find the N roots Find the eigenvectors Spectrum Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: We get a Nth polynomial in l: characteristic equation

  29. Position - momentum Quantum mechanics The uncertainty principle Finding a relationship between standard deviations for a pair of observables Uncertainty applies only for incompatible observables

  30. Derived from the Heisenberg’s equation of motion Special meaning of Dt Quantum mechanics The uncertainty principle Energy - time

  31. Quiz 33 Quantum mechanics Which one of these commutation relationships is not correct? A. B. C. D. E.

  32. z r y x The angular equation The radial equation Quantum mechanics Schrödinger equation in spherical coordinates

  33. Quantization of the energy Bohr radius Quantum mechanics The hydrogen atom

  34. E 0 E4 E3 Energy transition Paschen E2 Balmer E1 Rydberg constant Lyman Quantum mechanics The hydrogen atom Spectroscopy Energies levels

  35. z r y x Quantum mechanics The angular momentum eigenvectors Spherical harmonics are the eigenfunctions

  36. Quantum mechanics The spin

  37. Possible values for S when adding spins S1 and S2: Clebsch- Gordan coefficients Quantum mechanics Adding spins S

  38. Periodic table Quantum mechanics Filling the shells 2 2 6

  39. Periodic table Quantum mechanics

  40. Solids e- • Crystal Bloch’s theory • Free electron • gas theory Fermi surface Bravais k-space Quantum mechanics

  41. Good luck for finals And Merry Christmas! Quantum mechanics Thank you for your participation!

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