eigenmodes of planar drums and physics nobel prizes
DESCRIPTION
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 PresentationTRANSCRIPT
-
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!