Chaos and Quantum Chaos 2021

Table of contents

    1. Announcements
    2. Contents
    3. Lecture videos
    4. Live video conference
    5. Exercises
    6. Homework

Announcements

Homework sheet quantum chaos available (first due date 21.6.)
Enrollment (required for upload of homework on OPAL)

Contents

Hamiltonian chaos: dynamical systems, KAM-theorem, origin of chaotic dynamics, transport in phase space.
Quantum signatures of chaos: spectra, eigenstates, semiclassics, Gutzwiller trace formula, random matrix theory, experimental systems.
Computer simulations will be used for visualization.
Language: english
Knowledge: mechanics, basic quantum theory (Quantentheorie I), basic programming
Audience: Bachelor, Master (Vertiefung Theoretische Physik), Ph.D., IMPRS-MPSSE

Lecture videos

Videos for the week will be available by Wednesday:
Login at videocampus.sachsen required: click link, login, click link again
(in case of access problems, please contact me)

week videos (minutes) notes (.pdf)
1

0. Opening (5)
1.1 Motivation: What is chaos? (39)
1.2 What is quantum chaos? (23)
1.3 Contents (9)
2. Dynamical systems (24)
2.1 Linear dynamical systems (30)

0. Opening
1. Motivation
2.+2.1 Dynamical systems
2 2.2 Hamiltonian systems (49)
2.3 Integrable systems (40)
2.4 Linear maps - Cat map (69)

2.2+2.3 Dynamical systems
2.4 Dynamical systems
3 3.1 Hartmann-Grobmann theorem (28)
3.2 Poincaré section and map (11)
3.3 Billiards (28)
3.4 Time-periodic systems (63)		 3.1-3.3 Nonlinear systems
3.4 Nonlinear systems
4 3.5.1 KAM theorem:
Rational approximants (32)
3.5.2 Nonintegrable perturbation (44)
3.6 Poincaré-Birkhoff theorem (51)		 3.5 Nonlinear systems
3.6 Nonlinear systems
5

4.1 Chaotic dynamics: Where? (10)
4.2 Homoclinic tangle (31)
4.3 Melnikov method (1)
4.4a Smale's horseshoe map (37)
4.4b Symbolic dynamics (30)

4.1-4.3 Chaotic dynamics
4.4 Chaotic dynamics
6

4.5 Statistical concepts (11)
4.5.1 Ergodicity (25)
4.5.2 Mixing (10)
4.5.3 Kolmogorov-Sinai entropy (13)
4.5.4 Lyapunov exponents (13)
4.5.5 Shadowing theorem (7)

4.5-4.5.2 Chaotic dynamics
4.5.3-4.5.5 Chaotic dynamics
Pentecost break 24.-28.5.
7

5.1 Chaotic transport: General (22)
5.2 Partial transport barriers (40)
5.3 Average trapping time (6)
5.4 Distribution of trapping times (28)
5.5 Diffusion and anomalous diffusion (18)
5.6 Hamiltonian ratchets (23)

5.1-5.3 Chaotic transport
5.4-5.6 Chaotic transport
8

repeat:
1.2 What is quantum chaos? (23)
6.1 Chaotic dynamics in qm.? (51)
6.2 Time reversal (9)
6.3 Sensitive dependence on H (2)
6.4 Overview (6)
7.1 Eigenvalue statistics (12)
7.2 Level crossings (12)
7.3 Level spacing distribution (20)

6. Quantum chaos - intro
7.1-7.3 Eigenvalue statistics
9

7.4 Random matrix theory (56)
7.5 Average level density (35)
7.6 Unfolding of spectra (11)
7.7 Further level statistics (8)

7.4 Eigenvalue statistics
7.5-7.7 Eigenvalue statistics
10

8.1 Quantum billiards (11)
8.2 Real space properties (34)
8.3 Phase space representations (21)
(audio problems in 8.3)
8.4 Phase space properties (49)

8.1-8.2 Eigenfunctions
8.3-8.4 Eigenfunctions
11

9.1 Floquet theory (33)
9.2 Quantum kicked rotor:
9.2.1 Time-evolution (13)
9.2.2 Momentum basis (44)
9.2.3 Numerical considerations (17)
9.2.4 Comparison with classical (14)
9.3 Dynamical localization (24)

9. Time periodic q.m.
12

10. Semiclassics:
10.1 Integrable systems (15)
10.2 Propagator, density of states (24)
10.3 Semiclassical propagator (46)
10.4 Semiclassical Green function (12)
10.5 Gutzwiller trace formula (25)

10.1-10.3 Semiclassics
10.4-10.5 Semiclassics
13

10.6 Spectral statistics (45)
11. Chaotic Scattering:
11.1 Classical chaotic scattering (14)
11.2 S-matrix fluctuations (33)

10.6 Semiclassics
11.1-11.2 Scattering
14

11.3 Resonance states (50)
12. Experiments (12)

 11.3 Scattering

Live video conference

Every Monday 13:00 for (on average) 45 minutes we want to interact, starting 19.4.:
- questions and answers about lecture
- discussion of exercises/homework
BigBlueButton (Mondays, 13:00)   (for external participants without login)

Exercises

Mainly analytical problems on classical and quantum dynamical systems.
Will be presented by students in live video conference.
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Monday 7.6.: Homework H3
Monday 14.6.: Discussion on "chaos in quantum mechanics"
Monday 21.6.: Homework H4
Monday 28.6.: Discussion on "regular/chaotic eigenfunctions"
Monday 5.7.: Discussion on "Floquet states, quasienergies"
Monday 12.7.: Homework H5, H6
Monday 19.7.: Discussion on "Semiclassics"

Homework

Mainly numerical problems on classical and quantum dynamical systems.
Requirement for passing the class: 2 (out of 3) classical and 2 (out of 3) quantum problems with individual deadlines during the term. Presentation by students in live video conference after deadline is optional, but highly encouraged.
(Bachelor und Master: entspricht Schriftlicher Problembearbeitung).
Upload via OPAL
Homework classical chaos (due dates 10.5., 17.5., 7.6.)
Homework quantum chaos (due dates 21.6., 12.7., 12.7.)

About this page

Roland Ketzmerick
Last modified: Apr 03, 2022