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Physics 451

Dive deep into the intricate world of quantum mechanics and solid state physics with this comprehensive review. Covering topics such as free electron gas, Fermi surfaces, band structures, and the hydrogen atom, this guide will help you master key concepts for exams. Prepare for your final with confidence and good luck!

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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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