Eigenmodes of planar drums and Physics Nobel Prizes

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 Fa
rnham
Barnett Cohen Heller Lepore Vergini Saraceno
Clebsch Pockels Poisson Rayleigh Lamé Schwarz Webe
r
Banjai Betcke Descloux Driscoll Karageorghis Tolle
y Trefethen
Donnelly Eisenstat Fox Henrici Mason Moler Shryer
Benguria Exner Kuttler Levitin Sigillito
Berry Gutzwiller Keating Simon Wilkinson
and many more.
4

• 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

PLAN OF THE TALK
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
Schrödingers 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
Weyls 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

• Microwave experimentsSridhar Kudrolli 1994

Holographic interferometryMark Zarins 2004
Computations with DTD methodDriscoll 1997
20
(No Transcript)
21

• A MATHEMATICAL/NUMERICAL ASIDEWHICH 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
22
Michelson 1907
spectroscopy
also Bohr 1922, Bloembergen 1981
23

• 4. Line splitting

24
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
25
(?? 1.3)
Now halve the width of the connector.
(?? 0.017)
26
(?? 1.3)
(?? 4.1)
(?? 0.005)
27
Zeeman 1902
Stark 1919
Zeeman Stark effectsline splitting in
magnetic electric fields
also Michelson 1907, Dirac 1933, Lamb Kusch 1955
28

• 5. Localization

29
What if we make the bongo drums a little
asymmetrical?
? 0.2
?
?
? 0.2
30
(No Transcript)
31
(No Transcript)
32
GASKET WITH FOURFOLD SYMMETRY
Now we willshrink this holea little bit
33
GASKET WITH BROKEN SYMMETRY
34
Anderson 1977
Anderson localization and disordered materials
35

• 6. Resonance

36
A square cavitycoupled to a semi-infinite
stripof width 1
slightly abovethe minimumfrequency ?2
?
The spectrum is continuous ?2,?)
Close to theresonantfrequency ?? 2?2
?
37
Lengthening the slit strengthens the resonance
38
Marconi 1909
Purcell 1952
Radio
Nuclear Magnetic Resonance
also Bloch 1952
39
Townes 1964
Masers and lasers
Schawlow 1981
also Basov and Prokhorov 1964
40

• 7. Bound states

41
(No Transcript)
42
See papers by Exner and 1999 book by Londergan,
Carini Murdock.
43
Marie Curie 1903
Radioactivity
also Becquerel and Pierre Curie 1903 Rutherford
1908 Chemistry Marie Curie 1911
Chemistry Iréne Curie Joliot 1935 Chemistry
44
8. Eigenvalue avoidance
45
(No Transcript)
46
(No Transcript)
47
Another example of eigenvalue avoidance
48
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?
49
Weve mentioned
We might have added
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
Barkla 1917Compton 1927Raman 1930Stern
1943Pauli 1945Mössbauer 1961Landau
1962Kastler 1966Alfvén 1970Bardeen, Cooper
Schrieffer 1972Cronin Fitch 1980Siegbahn
1981von Klitzing 1985Ramsay 1989Brockhouse
1994Laughlin, Störmer Tsui 1998
50
Thank you!