Chaos in classical and quantum mechanics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1990
|
| Schriftenreihe: | Interdisciplinary applied mathematics
1 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 432 S. graph. Darst. |
| ISBN: | 0387971734 3540971734 |
Internformat
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| 245 | 1 | 0 | |a Chaos in classical and quantum mechanics |c Martin C. Gutzwiller |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1990 | |
| 300 | |a XIII, 432 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
| _version_ | 1804118445071007744 |
|---|---|
| adam_text | Contents
Preface
...........................................................................
vii
Introduction
.................................................................... 1
1.
The Mechanics of
Lagrange....................................... 5
. 1
Newton s Equations According to
Lagrange ................. 6
.2
The Variational Principle of
Lagrange......................... 7
.3
Conservation of Energy
.......................................... 10
.4
Example: Space Travel in a Given Time Interval;
Lambert s Formula
................................................ 11
.5
The Second Variation
............................................. 14
.6
The Spreading Trajectories
...................................... 16
2.
The Mechanics of Hamilton and Jacobi
........................ 19
2.1
Phase Space and Its Hamiltonian
............................... 19
2.2
The Action Function
S
........................................... 20
2.3
The Variational Principle of
Euler
and Maupertuis
......... 22
2.4
The Density of Trajectories on the Energy Surface
......... 23
2.5
Example: Space Travel with a Given Energy
................ 26
3.
Integrable
Systems
.................................................... 30
3.1
Constants of Motion and
Poisson
Brackets
................... 30
3.2
Invariant Tori and Action-Angle Variables
................... 32
3.3
Multiperiodic Motion
............................................. 34
3.4
The Hydrogen Molecule Ion
.................................... 36
3.5
Geodesies on
a Triaxial
Ellipsoid
.............................. 38
3.6
The
Toda
Lattice
.................................................. 41
3.7
Integrable
versus Separable
...................................... 43
; Contents
4.
The Three-Body Problem: Moon-Earth-Sun
................. 45
4.1
Reduction to Four Degrees of Freedom
....................... 45
4.2
Applications in Atomic Physics and Chemistry
.............. 48
4.3
The Action-Angle Variables in the Lunar Observations....
50
4.4
The Best Temporary Fit to a Kepler Ellipse
.................. 53
4.5
The Time-Dependent Hamiltonian
............................. 56
5.
Three Methods of Solution
......................................... 58
5.1
Variation of the Constants
(Lagrange)......................... 58
5.2
Canonical Transformations (Delaunay)
........................ 59
5.3
The Application of Canonical Transformations
.............. 62
5.4
Small Denominators and Other Difficulties
................... 63
5.5
Hill s Periodic Orbit in the Three-Body Problem
............ 65
5.6
The Motion of the Perigee and the Node
..................... 70
5.7
Displacements from the Periodic Orbit and Hill s Equation
72
6.
Periodic Orbits
......................................................... 75
6.1
Potentials with Circular Symmetry
............................. 77
6.2
The Number of Periodic Orbits in an
Integrable
System
... 80
6.3
The Neighborhood of a Periodic Orbit
........................ 82
6.4
Elliptic, Parabolic, and Hyperbolic Periodic Orbits
......... 84
7.
The Surface of Section
.............................................. 87
7.1
The Invariant Two-Form
......................................... 87
7.2
Integral Invariants and Liouville
s
Theorem
.................. 89
7.3
Area Conservation on the Surface of Section
................ 91
7.4
The Theorem of Darboux
........................................ 93
7.5
The Conjugation of Time and Energy in Phase Space
...... 95
8.
Models of the Galaxy and of Small Molecules
............... 99
8.1
Stellar Trajectories in the Galaxy
...............................100
8.2
The
Hénon-Heiles
Potential
.....................................102
8.3
Numerical Investigations
.........................................103
8.4
Some Analytic Results
...........................................106
8.5
Searching for Integrability with Kowalevskaya and
Painlevé
.............................................................109
8.6
Discrete Area-Preserving Maps
.................................
Ill
9.
Soft Chaos and the
KAM
Theorem
..............................116
9.1
The Origin of Soft Chaos
........................................116
9.2
Resonances in Celestial Mechanics
............................118
Contents xi
9.3
The Analogy with the Ordinary Pendulum
...................120
9.4
Islands of Stability and Overlapping Resonances
............125
9.5
How Rational Are the Irrational Numbers?
...................129
9.6
The
KAM
Theorem
...............................................132
9.7
Homoclinic Points
.................................................135
9.8
The Lore of the Golden Mean
..................................138
10.
Entropy and Other Measures of Chaos
.........................142
10.1
Abstact Dynamical Systems
....................................143
10.2
Ergodicity, Mixing, and K-Systems
..........................145
10.3
The Metric Entropy
..............................................147
10.4
The Automorphisms of the Torus
.............................149
10.5
The Topological Entropy
.......................................151
10.6
Anosov Systems and Hard Chaos
.............................154
11.
The
Anisotropie
Kepler Problem
.................................156
11.1
The Donor Impurity in a Semiconductor Crystal
...........156
11.2
Normalized Coordinates in the
Anisotropie
Kepler
Problem
............................................................159
11.3
The Surface of Section
..........................................161
11.4
Construction of Stable and Unstable Manifolds
............164
11.5
The Periodic Orbits in the
Anisotropie
Kepler Problem
..168
11.6
Some Questions Concerning the
АКР
.......................171
12.
The Transition from Classical to Quantum Mechanics
.....173
12.1
Are Classical Mechanics and Quantum Mechanics
Compatible?
....................................................... 174
12.2
Changing Coordinates in the Action
..........................176
12.3
Adding Actions and Multiplying Probabilities
..............178
12.4
Rutherford Scattering
............................................180
12.5
The Classical Version of Quantum Mechanics
.............184
12.6
The Propagator in Momentum Space
.........................186
12.7
The Classical Green s Function
...............................188
12.8
The Hydrogen Atom in Momentum Space
..................190
13.
The New World of Quantum Mechanics
.......................194
13.1
The Liberation from Classical Chaos
.........................194
13.2
The Time-Dependent
Schrödinger
Equation
................196
13.3
The Stationary
Schrödinger
Equation
........................198
13.4
Feynman s Path Integral
........................................200
13.5
Changing Coordinates in the Path Integral
..................202
13.6
The Classical Limit
..............................................204
xii Contents
14.
The Quantization of
Integrable
Systems
........................207
14.1
Einstein s Picture of Bohr s Quantization Rules
...........208
14.2
Keller s Construction of Wave Functions and Maslov
Indices
.............................................................211
14.3
Transformation to Normal Forms
.............................215
14.4
The Frequency Analysis of a Classical Trajectory
.........220
14.5
The Adiabatic Principle
.........................................224
14.6
Tunneling Between Tori
........................................227
15.
Wave Functions in Classically Chaotic Systems
.............231
15.1
The Eigenstates of an
Integrable
System
....................232
15.2
Patterns of Nodal Lines
.........................................233
15.3
Wave-Packet Dynamics
.........................................238
15.4
Wigner s Distribution Function in Phase Space
............241
15.5
Correlation Lengths in Chaotic Wave Functions
...........247
15.6
Scars, or What Is Left of the Classical Periodic Orbits...
249
16.
The Energy Spectrum of a Classically Chaotic System
....254
16.1
The Spectrum as a Set of Numbers
...........................255
16.2
The Density of States and Weyl s Formula
.................257
16.3
Measures for Spectral Fluctuations
...........................261
16.4
The Spectrum of Random Matrices
...........................263
16.5
The Density of States and Periodic Orbits
..................266
16.6
Level Clustering in the Regular Spectrum
...................270
16.7
The Fluctuations in the Irregular Spectrum
.................273
16.8
The Transition from the Regular to the Irregular Spectrum
275
16.9
Classical Chaos and Quantal Random Matrices
............278
17.
The Trace Formula
...................................................282
17.1
The Van Vleck Formula Revisited
............................283
17.2
The Classical Green s Function in Action-Angle
Variables
..........................................................285
17.3
The Trace Formula for
Integrable
Systems
..................287
17.4
The Trace Formula in Chaotic Dynamical Systems
.......291
17.5
The Mathematical Foundations of the Trace Formula
.....295
17.6
Extensions and Applications
...................................298
17.7
Sum Rules and Correlations
...................................301
17.8
Homogeneous Hamiltonians
...................................305
17.9
The Riemann Zeta-Function
...................................307
17.10
Discrete Symmetries and the
Anisotropie
Kepler Problem
312
17.11
From Periodic Orbits to Code Words
.......................314
17.12
Transfer Matrices
...............................................317
Contents xiii
18.
The Diamagnetic Kepler Problem
................................322
18.1
The Hamiltonian in the Magnetic Field
......................323
18.2
Weak Magnetic Fields and the Third Integral
..............325
18.3
Strong Fields and Landau Levels
.............................326
18.4
Scaling the Energy and the Magnetic Field
.................329
18.5
Calculation of the Oscillator Strengths
.......................332
18.6
The Chaotic Spectrum in Terms of Closed Orbits
.........336
19.
Motion on a Surface of Constant Negative Curvature
......340
19.1
Mechanics in a Riemannian Space
............................341
19.2
Poincaré s
Model of Hyperbolic Geometry
.................345
19.3
The Construction of Polygons and Tilings
..................348
19.4
The Geodesies on a Double Torus
............................354
19.5
Selberg s Trace Formula
........................................358
19.6
Computations on the Double Torus
...........................363
19.7
Surfaces in Contact with the Outside World
................369
19.8
Scattering on a Surface of Constant Negative Curvature
.374
19.9
Chaos in Quantum-Mechanical Scattering
...................377
19.10
The Classical Interpretation of the Quantal Scattering
...379
20.
Scattering Problems, Coding, and Multifractal Invariant
Measures
................................................................383
20.1
Electron Scattering in a Muffin-Tin Potential
...............384
20.2
The Coding of Geodesies on a Singular Polygon
..........389
20.3
The Geometry of the Continued Fractions
..................393
20.4
A New Measure in Phase Space Based on the Coding
...395
20.5
Invariant Multifractal Measures in Phase Space
............398
20.6
Multifractals in the
Anisotropie
Kepler Problem
...........402
20.7
Bundling versus Pruning a Binary Tree
......................407
References
......................................................................410
Index
.............................................................................427
|
| any_adam_object | 1 |
| author | Gutzwiller, Martin C. |
| author_facet | Gutzwiller, Martin C. |
| author_role | aut |
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| dewey-ones | 530 - Physics |
| dewey-raw | 530.12 |
| dewey-search | 530.12 |
| dewey-sort | 3530.12 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| format | Book |
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| genre | (DE-588)1071861417 Konferenzschrift gnd-content Matériel didactique |
| genre_facet | Konferenzschrift Matériel didactique |
| id | DE-604.BV004248737 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:10:24Z |
| institution | BVB |
| isbn | 0387971734 3540971734 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002643170 |
| oclc_num | 300912857 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-91 DE-BY-TUM DE-384 DE-12 DE-19 DE-BY-UBM DE-739 DE-355 DE-BY-UBR DE-20 DE-29T DE-703 DE-634 DE-83 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-91 DE-BY-TUM DE-384 DE-12 DE-19 DE-BY-UBM DE-739 DE-355 DE-BY-UBR DE-20 DE-29T DE-703 DE-634 DE-83 DE-188 |
| physical | XIII, 432 S. graph. Darst. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Springer |
| record_format | marc |
| series | Interdisciplinary applied mathematics |
| series2 | Interdisciplinary applied mathematics |
| spelling | Gutzwiller, Martin C. Verfasser aut Chaos in classical and quantum mechanics Martin C. Gutzwiller New York [u.a.] Springer 1990 XIII, 432 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Interdisciplinary applied mathematics 1 Astronomie Chaos (théorie des systèmes) ram Chaos - Théorie quantique Chaos déterministe Jussieu Chaos obspm Mécanique Mécanique classique obspm Mécanique céleste obspm Mécanique quantique obspm Mécanique ram Théorie quantique ram Quantenchaos (DE-588)4130849-9 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Mechanik (DE-588)4038168-7 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Matériel didactique Mechanik (DE-588)4038168-7 s Chaos (DE-588)4191419-3 s DE-604 Quantenchaos (DE-588)4130849-9 s Chaostheorie (DE-588)4009754-7 s Theoretische Mechanik (DE-588)4185100-6 s Quantenmechanik (DE-588)4047989-4 s Interdisciplinary applied mathematics 1 (DE-604)BV004216726 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002643170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gutzwiller, Martin C. Chaos in classical and quantum mechanics Interdisciplinary applied mathematics Astronomie Chaos (théorie des systèmes) ram Chaos - Théorie quantique Chaos déterministe Jussieu Chaos obspm Mécanique Mécanique classique obspm Mécanique céleste obspm Mécanique quantique obspm Mécanique ram Théorie quantique ram Quantenchaos (DE-588)4130849-9 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Chaostheorie (DE-588)4009754-7 gnd Chaos (DE-588)4191419-3 gnd Quantenmechanik (DE-588)4047989-4 gnd Mechanik (DE-588)4038168-7 gnd |
| subject_GND | (DE-588)4130849-9 (DE-588)4185100-6 (DE-588)4009754-7 (DE-588)4191419-3 (DE-588)4047989-4 (DE-588)4038168-7 (DE-588)1071861417 |
| title | Chaos in classical and quantum mechanics |
| title_auth | Chaos in classical and quantum mechanics |
| title_exact_search | Chaos in classical and quantum mechanics |
| title_full | Chaos in classical and quantum mechanics Martin C. Gutzwiller |
| title_fullStr | Chaos in classical and quantum mechanics Martin C. Gutzwiller |
| title_full_unstemmed | Chaos in classical and quantum mechanics Martin C. Gutzwiller |
| title_short | Chaos in classical and quantum mechanics |
| title_sort | chaos in classical and quantum mechanics |
| topic | Astronomie Chaos (théorie des systèmes) ram Chaos - Théorie quantique Chaos déterministe Jussieu Chaos obspm Mécanique Mécanique classique obspm Mécanique céleste obspm Mécanique quantique obspm Mécanique ram Théorie quantique ram Quantenchaos (DE-588)4130849-9 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Chaostheorie (DE-588)4009754-7 gnd Chaos (DE-588)4191419-3 gnd Quantenmechanik (DE-588)4047989-4 gnd Mechanik (DE-588)4038168-7 gnd |
| topic_facet | Astronomie Chaos (théorie des systèmes) Chaos - Théorie quantique Chaos déterministe Chaos Mécanique Mécanique classique Mécanique céleste Mécanique quantique Théorie quantique Quantenchaos Theoretische Mechanik Chaostheorie Quantenmechanik Mechanik Konferenzschrift Matériel didactique |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002643170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004216726 |
| work_keys_str_mv | AT gutzwillermartinc chaosinclassicalandquantummechanics |