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Eigenmodes of planar drums and Physics Nobel Prizes

Eigenmodes of planar drums and Physics Nobel Prizes. Nick Trefethen Numerical Analysis Group Oxford University Computing Laboratory. Consider an idealized “drum”: Laplace operator in 2D region with zero boundary conditions.

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Eigenmodes of planar drums and Physics Nobel Prizes

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  1. Eigenmodes of planar drumsand Physics Nobel Prizes Nick TrefethenNumerical Analysis GroupOxford University Computing Laboratory

  2. Consider an idealized “drum”: Laplace operator in 2D region with zero boundary conditions. Basis of the presentation: numerical methods developedwith Timo Betcke, U. of Manchester B. & T., “Reviving the method of particularsolutions,” SIAM Review, 2005 T. & B., “Computed eigenmodes of planarregions,” Contemporary Mathematics, 2006

  3. Goldstone Jaffe Lenz Ravenhall Schult Wyld Contributors to this subject include Bäcker Boasmann Riddel Smilansky Steiner Conway Farnham Barnett Cohen Heller Lepore Vergini Saraceno Clebsch Pockels Poisson Rayleigh Lamé Schwarz Weber Banjai Betcke Descloux Driscoll Karageorghis Tolley Trefethen Donnelly Eisenstat Fox Henrici Mason Moler Shryer Benguria Exner Kuttler Levitin Sigillito Berry Gutzwiller Keating Simon Wilkinson and many more.

  4. PLAN OF THE TALK EIGHT EXAMPLES 1. L-shaped region 2. Higher eigenmodes 3. Isospectral drums 4. Line splitting 5. Localization 6. Resonance 7. Bound states 8. Eigenvalue avoidance [Numerics]  Eigenmodes of drums Physics Nobel Prizes

  5. 1. L-shaped region Fox-Henrici-Moler 1967

  6. Mathematicallycorrect alternative: Cleve Moler, late 1970s

  7. Some higher modes, to 8 digits tripledegeneracy: 52 + 52 = 12 + 72 = 72 + 12

  8. Circular L shape no degeneracies,so far as I know

  9. Schrödinger 1933 Schrödinger’s equation andquantum mechanics also Heisenberg 1932, Dirac 1933

  10. 2. Higher eigenmodes

  11. Eigenmodes 1-6

  12. Eigenmodes 7-12

  13. Eigenmodes 13-18

  14. Six consecutive higher modes

  15. Weyl’s Law n~ 4n / A ( A = area of region )

  16. Planck 1918 blackbody radiation also Wien 1911

  17. 3. Isospectral drums

  18. Kac 1966 “Can one hear the shape of a drum?” Gordon, Webb & Wolpert 1992 “Isospectral plane domains and surfaces via Riemannian orbifolds” Numerical computations: Driscoll 1997 “Eigenmodes of isospectral drums”

  19. Holographic interferometryMark Zarins 2004 Microwave experimentsSridhar & Kudrolli 1994 Computations with “DTD method”Driscoll 1997

  20. A MATHEMATICAL/NUMERICAL ASIDE:WHICH CORNERS CAUSE SINGULARITIES? That is, at which corners are eigenfunctions not analytic?(Effective numerical methods need to know this.) Answer: all whose angles are / ,   integer Proof: repeated analytic continuation by reflection E.G., the corners marked inred are the singular ones:

  21. Michelson 1907 spectroscopy also Bohr 1922, Bloembergen 1981

  22. 4. Line splitting

  23. “Bongo drums”—two chambers, weakly connected.     Without the coupling the eigenvalues are { j 2 + k 2 } = { 2, 2, 5, 5, 5, 5, 8, 8, 10, 10, 10, 10, … } With the coupling…

  24. ( = 1.3%) Now halve the width of the connector. ( =0.017%)

  25. ( =1.3%) ( =4.1%) ( =0.005%)

  26. Stark 1919 Zeeman 1902 Zeeman & Stark effects:line splitting in magnetic & electric fields also Michelson 1907, Dirac 1933, Lamb & Kusch 1955

  27. 5. Localization

  28. What if we make the bongo drums a little asymmetrical?   + 0.2   + 0.2

  29. GASKET WITH FOURFOLD SYMMETRY Now we willshrink this holea little bit

  30. GASKET WITH BROKEN SYMMETRY

  31. Anderson 1977 Anderson localization and disordered materials

  32. 6. Resonance

  33. A square cavitycoupled to a semi-infinite stripof width 1 slightly abovethe minimumfrequency 2  The spectrum is continuous: [2,) Close to theresonantfrequency  22 

  34. Lengthening the slit strengthens the resonance:

  35. Marconi 1909 Purcell 1952 Nuclear Magnetic Resonance Radio also Bloch 1952

  36. Masers and lasers Townes 1964 Schawlow 1981 also Basov and Prokhorov 1964

  37. 7. Bound states

  38. 8.66503 9.17005

  39. See papers by Exner and 1999 book by Londergan, Carini & Murdock.

  40. Marie Curie 1903 Radioactivity also Becquerel and Pierre Curie 1903 Rutherford 1908 [Chemistry] Marie Curie 1911 [Chemistry] Iréne Curie & Joliot 1935 [Chemistry]

  41. 8. Eigenvalue avoidance

  42. Another example of eigenvalue avoidance

  43. Wigner 1963 Energy states of heavy nucleieigenvalues of random matrices cf. Montgomery, Odlyzko, … “The 1020th zero of the Riemannzeta function and 70 million of its neighbors” … Riemann Hypothesis?…

  44. We might have added: We’ve mentioned: Barkla 1917Compton 1927Raman 1930Stern 1943Pauli 1945Mössbauer 1961Landau 1962Kastler 1966Alfvén 1970Bardeen, Cooper & Schrieffer 1972Cronin & Fitch 1980Siegbahn 1981von Klitzing 1985Ramsay 1989Brockhouse 1994Laughlin, Störmer & Tsui 1998 … Zeeman 1902Becquerel, Curie & Curie 1903Michelson 1907Marconi & Braun 1909Wien 1911Planck 1918Stark 1919Bohr 1922Heisenberg 1932Dirac 1933Schrödinger 1933Bloch & Purcell 1952Lamb & Kusch 1955Wigner 1963Townes, Basov & Prokhorov 1964Anderson 1977Bloembergen & Schawlow 1981

  45. Thank you!

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