eigenmodes of planar drums and physics nobel prizes nick trefethen numerical analysis group oxford...
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Eigenmodes of planar drumsand Physics Nobel Prizes
Nick TrefethenNumerical Analysis Group
Oxford University Computing Laboratory
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Consider an idealized “drum”:Laplace operator in 2D regionwith 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
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Contributors to this subject include
BarnettCohenHellerLeporeVerginiSaraceno
BanjaiBetckeDesclouxDriscollKarageorghisTolleyTrefethen
ClebschPockelsPoissonRayleighLaméSchwarzWeber
GoldstoneJaffeLenzRavenhallSchultWyld
DonnellyEisenstatFoxHenriciMasonMolerShryer
BenguriaExnerKuttlerLevitinSigillito
BerryGutzwillerKeatingSimonWilkinson
BäckerBoasmannRiddelSmilanskySteinerConwayFarnham
and many more.
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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
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1. L-shaped region
Fox-Henrici-Moler 1967
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Cleve Moler, late 1970s
Mathematicallycorrect alternative:
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Some higher modes, to 8 digits
tripledegeneracy:
52 + 52 =12 + 72 =72 + 12
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Circular L shape
no degeneracies,so far as I know
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Schrödinger 1933
Schrödinger’s equation andquantum mechanics
also Heisenberg 1932, Dirac 1933
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2. Higher eigenmodes
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Eigenmodes 1-6
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Eigenmodes 7-12
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Eigenmodes 13-18
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Six consecutive higher modes
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Weyl’s Law
n ~ 4n / A
( A = area of region )
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Planck 1918
blackbody radiation
also Wien 1911
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3. Isospectral drums
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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”
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Microwave experimentsSridhar & Kudrolli 1994
Computations with “DTD method”Driscoll 1997
Holographic interferometryMark Zarins 2004
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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:
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Michelson 1907
spectroscopy
also Bohr 1922, Bloembergen 1981
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4. Line splitting
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“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…
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Now halve the width of the connector.
( = 1.3%)
( = 0.017%)
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( = 1.3%)
( = 4.1%)
( = 0.005%)
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Zeeman 1902
Zeeman & Stark effects:line splitting in magnetic & electric fields
also Michelson 1907, Dirac 1933, Lamb & Kusch 1955
Stark 1919
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5. Localization
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What if we make the bongo drums a little asymmetrical?
+ 0.2
+ 0.2
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GASKET WITH FOURFOLD SYMMETRY
Now we w
ill
shrin
k this
hole
a little
bit
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GASKET WITH BROKEN SYMMETRY
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Anderson 1977
Anderson localizationand disordered materials
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6. Resonance
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A square cavitycoupled to a
semi-infinite stripof width 1
slightly abovethe minimumfrequency 2
The spectrum iscontinuous: [2,)
Close to theresonantfrequency 22
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Lengthening the slit strengthens the resonance:
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Marconi 1909
also Bloch 1952
Purcell 1952
Radio Nuclear Magnetic Resonance
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Townes 1964
Masers and lasers
also Basov and Prokhorov 1964
Schawlow 1981
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7. Bound states
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9.17005
8.66503
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See papers by Exner and 1999 book by Londergan, Carini & Murdock.
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Marie Curie 1903
Radioactivity
also Becquerel and Pierre Curie 1903 Rutherford 1908 [Chemistry]Marie Curie 1911 [Chemistry]Iréne Curie & Joliot 1935 [Chemistry]
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8. Eigenvalue avoidance
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Another example of eigenvalue avoidance
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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?…
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We’ve mentioned: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
We might have added:
Barkla 1917Compton 1927Raman 1930Stern 1943Pauli 1945Mössbauer 1961Landau 1962Kastler 1966Alfvén 1970Bardeen, Cooper & Schrieffer 1972Cronin & Fitch 1980Siegbahn 1981von Klitzing 1985Ramsay 1989Brockhouse 1994Laughlin, Störmer & Tsui 1998 …
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Thank you!