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 lecture 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
    

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Lecture Notes:


    

  1. Class from Oct. 13th
    2. 

  2. Class from Oct. 17th
    3. 

  3. Class from Oct. 21st
    4. 

  4. Class from Oct. 24th
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  5. Class from Oct. 27th
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  6. Class from Oct. 31st
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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.
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  2. Sheet to be discussed on Tue, Nov. 4th 
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  3. Sheet to e discussed on Tue, Nov 11th
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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.