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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. - PowerPoint PPT Presentation
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Eigenmodes of planar drumsand Physics Nobel PrizesNick TrefethenNumerical Analysis GroupOxford University Computing Laboratory
Consider an idealized drum:Laplace operator in 2D regionwith zero boundary conditions.Basis of the presentation: numerical methods developed with Timo Betcke, U. of ManchesterB. & T., Reviving the method of particular solutions, SIAM Review, 2005
T. & B., Computed eigenmodes of planar regions, Contemporary Mathematics, 2006
Contributors to this subject includeBarnettCohenHellerLeporeVerginiSaraceno
BanjaiBetckeDesclouxDriscollKarageorghisTolleyTrefethen
ClebschPockelsPoissonRayleighLamSchwarzWeber
GoldstoneJaffeLenzRavenhallSchultWyldDonnellyEisenstatFoxHenriciMasonMolerShryerBenguriaExnerKuttlerLevitinSigillitoBerryGutzwillerKeatingSimonWilkinsonBckerBoasmannRiddelSmilanskySteinerConwayFarnhamand many more.
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
PLAN OF THE TALK
1. L-shaped regionFox-Henrici-Moler 1967
Cleve Moler, late 1970sMathematically correct alternative:
Some higher modes, to 8 digitstriple degeneracy: 52 + 52 =12 + 72 =72 + 12
Circular L shapeno degeneracies, so far as I know
Schrdinger 1933Schrdingers equation and quantum mechanicsalso Heisenberg 1932, Dirac 1933
2. Higher eigenmodes
Eigenmodes 1-6
Eigenmodes 7-12
Eigenmodes 13-18
Six consecutive higher modes
Weyls Lawn ~ 4n / A( A = area of region )
Planck 1918blackbody radiationalso Wien 1911
3. Isospectral drums
Kac 1966Can one hear the shape of a drum? Gordon, Webb & Wolpert 1992Isospectral plane domains and surfaces via Riemannian orbifolds
Numerical computations:Driscoll 1997Eigenmodes of isospectral drums
Microwave experiments Sridhar & Kudrolli 1994 Computations with DTD method Driscoll 1997 Holographic interferometry Mark Zarins 2004
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 / , integerProof: repeated analytic continuation by reflection
E.G., the corners marked in red are the singular ones:
Michelson 1907spectroscopyalso Bohr 1922, Bloembergen 1981
4. Line splitting
Bongo drumstwo 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
Now halve the width of the connector.( = 1.3%)( = 0.017%)
( = 1.3%)( = 4.1%)( = 0.005%)
Zeeman 1902Zeeman & Stark effects: line splitting in magnetic & electric fieldsalso Michelson 1907, Dirac 1933, Lamb & Kusch 1955Stark 1919
5. Localization
What if we make the bongo drums a little asymmetrical? + 0.2 + 0.2
GASKET WITH FOURFOLD SYMMETRYNow we will shrink this hole a little bit
GASKET WITH BROKEN SYMMETRY
Anderson 1977Anderson localizationand disordered materials
6. Resonance
A square cavity coupled to a semi-infinite strip of width 1slightly above the minimum frequency 2 The spectrum iscontinuous: [2,)Close to the resonant frequency 22
Lengthening the slit strengthens the resonance:
Marconi 1909also Bloch 1952Purcell 1952RadioNuclear Magnetic Resonance
Townes 1964Masers and lasersalso Basov and Prokhorov 1964Schawlow 1981
7. Bound states
See papers by Exner and 1999 book by Londergan, Carini & Murdock.
Marie Curie 1903Radioactivityalso Becquerel and Pierre Curie 1903 Rutherford 1908 [Chemistry]Marie Curie 1911 [Chemistry]Irne Curie & Joliot 1935 [Chemistry]
8. Eigenvalue avoidance
Another example of eigenvalue avoidance
Wigner 1963Energy states of heavy nuclei eigenvalues of random matrices
cf. Montgomery, Odlyzko, The 1020th zero of the Riemann zeta function and 70 million of its neighbors Riemann Hypothesis?
Weve mentioned:Zeeman 1902Becquerel, Curie & Curie 1903Michelson 1907Marconi & Braun 1909Wien 1911Planck 1918Stark 1919Bohr 1922Heisenberg 1932Dirac 1933Schrdinger 1933Bloch & Purcell 1952Lamb & Kusch 1955Wigner 1963Townes, Basov & Prokhorov 1964Anderson 1977Bloembergen & Schawlow 1981We might have added:Barkla 1917Compton 1927Raman 1930Stern 1943Pauli 1945Mssbauer 1961Landau 1962Kastler 1966Alfvn 1970Bardeen, Cooper & Schrieffer 1972Cronin & Fitch 1980Siegbahn 1981von Klitzing 1985Ramsay 1989Brockhouse 1994Laughlin, Strmer & Tsui 1998
Thank you!