Fachbereich Mathematik

Interacting Many-Body Systems

Content: The main focus of this class is the derivation and rigorous justification of many body quantum systems. 

We will discuss the difference between Bosons and Fermions, i.e. between symmetrized and anti-symmatrized wave-functions.

Most of the time we will consider interacting systems at low temperature. We will prove that the correlations caused by the interaction will be sufficiently mild such that law of large numbers arguments are still valid. Thus we can approximate the interaction by its expectation value and arrive at a non-linear evolution equation for the gas.

We will consider both the static and the dynamic case: We will show that the ground state of the system is well approximated by the ground state of the respective non-linear theory, and show that the dynamics of the gas is well approximated by the respective non-linear Schrödinger equation.

Later we will consider higher order corrections and, in particular, derive Bogoliubov-theory. 

Many interesting effects, for example, the speed of sound in a Bose gas will be discussed.

Time and place: The lectures take place

Mondays 10:15-12:00 in N14

Fridays 10:15-12:00 in N14

It is possible to attend the lecture in person or via Zoom: https://zoom.us/j/93415213988

All lectures will also be recorded, the videos of the lecture can be found below.

Recordings:

  1. Class from Oct. 13th: https://zoom.us/rec/share/Ey1OIRqvx9rZbXsup8LBfOEl1SVOYL36QmqrTW69ofRbqDIgRS_X8ayz19ul_ue8._hkKcCjnY1EQNm-S Passcode: .A@z2+N
  2. Class from Oct. 17th: https://zoom.us/rec/share/1CR31nzU3x1q8AF_q53Rzov-5zeEfGzxmax2BakX0NuEsq2XtZbvUfoJ5gjE81Ov.3_roJr0r83r8wjut  Passcode: B?JGb@s3 
  3. Class from Oct. 24th: https://zoom.us/rec/share/4SpzYGFzup9_1JSeZWBYvyErLeUVgovXhohi3FCTeV1J6gqZPOWpK0T7b_-gyY-h.HMXxUnaG7iHyfuoc Passcode: 3vc7%.cG
  4. Class from Oct. 27th: https://zoom.us/rec/share/2Zg4JZxfFRptLsQ-j64JkBMsk3CEHUm5_W1uZtz2Jv7PeELKPNfvsgOdHcN_zI9w.rC9IwVzfwkQbYiu0?startTime=1761556456000
    Passcode: nN1Cfz=n
  5. Class from Oct. 31st: https://zoom.us/rec/share/txTd-TlpjuRFds_a7soNve6Y5B7BfCZhDlmkENp3QmmHmjeERXkvONSWxCoTIX0G.OkqmUMEXMY4MW8AW Passcode: mt*8@Wpk
  6. Class from Nov. 3rd https://zoom.us/rec/share/ivTj1OvF7lvUskSs2GEq3Zu13Df33xBWtghVxQOkUAvL4FtELyLVwQZoOW4pgp01.k6Np4hyrl9kbCJXS Passcode: SUXD%8=Z
  7. Class from Nov 10th https://zoom.us/rec/share/xQtS6apWHY44K6jkrwDVWhOGfC8ppuJ8D7D4ahN6OdxxECBFsAF4CbKmnfENIL9B.s904M-x5zBOgXffx
    Passcode: 2j#j!z&P
  8. Passcode: zb*4EaUg
  9. Class from Nov. 17th:
    Passcode: 2T@$nd!p
  10. Class from Nov. 21st: https://zoom.us/rec/share/tWH-54n7V8p7l5PEmOb1bI8vybzl8XmvnjaddaGTUI4rCVpWIRFEBsUEQKOBFOa2.uGnSdhx5YxT7UKT6
    Passcode: $y6M8#Q4
  11. Class from Nov. 24th:
    Passcode: !B6+Ac1E
  12. Class from Nov. 28th
    Passcode: ^0q317xH
  13. Class from Dez. 1st:
    Passcode: p.XJyUp2
  14. Class from Dez. 5th:
    Passcode: 8R3&p#Rb
  15. Class from Dez. 8th:
    Passcode: .$vW=?P0

Lecture Notes:

  1. Class from Oct. 13th
  2. Class from Oct. 17th
  3. Class from Oct. 21st
  4. Class from Oct. 24th
  5. Class from Oct. 27th
  6. Class from Oct. 31st
  7. Class from Nov. 3rd
  8. Class from Nov. 7th
  9. Class from Nov. 10th
  10. Class form Nov 14th
  11. Class from Nov. 17th
  12. Class from Nov. 21st
  13. Class from Nov. 24th
  14. Class from Nov. 28th
  15. Class from Dez. 1st
  16. Class from Dez. 5th
  17. Class from Dez. 8th

Exercises: Each week I will upload an exercise sheet that we will discuss during exercise classes. Exercise classes take place on Tuesdays from 4:00 pm until 6 pm in C4H33.

  1. Sheet to be discussed on Tue, Oct. 28th.
  2. Sheet to be discussed on Tue, Nov. 4th
  3. Sheet to be discussed on Tue, Nov 11th
  4. Sheet to be discussed on Tue, Nov 18th
  5. Sheet to be discussed on Tue, Nov 25th
  6. Sheet to be discussed on Tue, Dez 2nd
  7. Sheet to be discussed on Tue, Dez. 9th  sketch of solutions
  8. Sheet to be sidcussed on Tue, Dez. 16th

Exam: There will be a written exam on Monday, Feb. 16th, from 9-11 am in N14.

Registration for the course: Please register for the course at URM using this link: https://urm.math.uni-tuebingen.de/student/anmelden?lv_id=1419.

Registration for the exams is mandatory, details will be explained in class.

Additional Info: The course has 9 ECTS. It counts as specialization in Mathematical Physics or Analysis.