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Periodic Oscillations: Ubiquitous or Merely a Paradigm?

Explore the mathematical description of physically and electronically coupled systems, showcasing how symmetry, oscillations, and quantum mechanics describe everyday phenomena. Includes problem sets, journal presentations, and a final exam.

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Periodic Oscillations: Ubiquitous or Merely a Paradigm?

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  1. PH427are periodic oscillations ubiquitous or merely just a paradigm? Paradigm: Periodic Systems Instructor: Matt Graham Winter 2016

  2. smaller goal: full mathematical description of physically & electronically coupled systemsBIG GOAL: show how symmetry, oscillations and quantum mechanics really describe “everyday stuff”i.e. quantum mechanics you can get paid to do!!

  3. 35%  problem sets (3)15%  pick a “solid state physics” journal article, give a 10 min. talk50%  final exam (M of exam week)

  4. Roadmap DAYS 1- 7: • Coupled pendulum, railroad cars, atoms, etc. • From atoms to crystals  extend coupling to infinity and define a dispersion relation for an atomic system DAYS 7- 15: • Quantum wavestates in periodic systems • 5 minute journal presentations

  5. Translational Symmetry and Noether’s Theorem Any system with translational symmetry has an associated momentum conservation law.  An electron moving through a perfectly periodic crystal maintains its momentum like the electron was travelling though a vaccuum Graphene

  6. Draw the “Unit Cell” Graphene

  7. Draw the “Unit Cell” Graphene

  8. Draw the “Unit Cell” Celtic knot Protein/DNA

  9. Draw the “Unit Cell” Celtic knot Protein/DNA

  10. ? ?

  11. PERIODIC SYSTEM IN BIOLOGY:Light Harvesting Complex II

  12. Bacterial Light Harvesting Hu, et al. J. Phys. Chem. B (1997) 101 3854 Bahatyrova, et al. Nature (2004) 430 1058

  13. “Beats” in Motion of Coupled Oscillator

  14. 1 2

  15. Chain of Many (N) Spring-Coupled Masses

  16. Two Masses Connected to Hooke’s Law Springs Force on m1: Force on m2:

  17. Low frequency mode: Symmetric mode: A1 = A2 High frequency mode: Antisymmetric mode: A1 = A2

  18. symmetric mode (low frequency) anti-symmetric mode (high frequency) From Fig. 8.3, I. G. Main, Vibrations and Waves in Physics

  19. 5 Mass Chain – Mode n=1 l = 12a k = 2p / l = p/ 6a

  20. 5 Mass Chain – Mode n=2 • = 6a k2 = 2p / l = p/ 3a • = 2k

  21. 5 Mass Chain – Mode n=3 • = 4a k3 = 2p / l = p/ 2a • = 3k

  22. 5 Mass Chain – Mode n=4 • = 3a k4 = 2p / l = 2p/ 3a • = 4k

  23. 5 Mass Chain – Mode 5 • = 12a/5  k5 = 2p / l = 5p/6a • = 5k

  24. 5 Mass Chain – Mode 6 • = 2a k6 = 2p / l = p/a • = 6k

  25. 5 Mass Chain – Mode 7 • = 12a/7  k7 = 2p / l = 7p/6a • = 7k

  26. 5 Mass Chain Dispersion Relation

  27. Diatomic chain, 16 masses, 8 unit cells

  28. Dispersion Relation for m =1, M = 2

  29. Dispersion Relation for m = 1, M = 1.1

  30. Amplitude (m)/Amplitude (M)  (m = 1, M = 2)

  31. Amplitude (m)/Amplitude (M)  (m = 1, M = 1.1)

  32. Damped Wave in Forbidden Frequency Range(Evanescent Wave)

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