PHYS625-0101: Non-relativistic Quantum Mechanics-Spring 2022 jaydsau

PHYS625-0101: Non-relativistic Quantum Mechanics-Spring 2022 jaydsau

 

An introduction to field theory in condensed matter physics:

Phys 625 introduces field theory techniques  for the understanding of many-body condensed matter systems focusing mostly on the many-electron system. While quantum field theory originated in high-energy physics, it also provides the framework to understand many-body quantum systems specifically response as well as perturbation theory. A key complication is that Lorentz invariance typically doesn't apply to condensed matter systems, but simplification is the presence of a natural regularization scale.  Familiarity with graduate level quantum mechanics, statistical mechanics and E&M will be assumed.

 

 

 

 

 

 

Some of the topics covered are (suggestions welcome):

  • Phonons as an intro to quantum fields
  • Field theory for fermions 
  • Jordan-Wigner/Hubbard models etc
  • Green functions, Wick's theorem etc
  • Linked cluster theorem, RPA etc
  • Fermi surface and Fermi liquids - zero sound
  • Imaginary time Green functions
  • Linear response, Kubo formula etc
  • Diagrams for electron diffusion versus localization
  • Landau theory of phase transitions
  • Goldstone theorem
  • Anderson-Higgs mechanism
  • Path integral for Bosons/Fermions
  • Coherent state path integral for magnets
  • Hubbard stratonovich transformation
  • Nambu-Gorkov Green function approach to superconductivity
  • Local moments and the Kondo effect 

Required Resources

Course website: elms.umd.edu

Text book: Introduction to Many-body physics, Piers Coleman, 1st edition, Cambridge University Press 2015.

 

       

 

 

 

                                           

 

Dr. Jay D. Sau

jaydsau@umd.edu

Class Meets

Mondays & Wednesdays

12:30 pm – 1:45 pm

Phy1219

Office Hours

PSC 3143 by appointment

 

Grader

TBD

TBD

Suggested Prerequisites

Phys 604, 606. 622, 623

Graduate E&M, quantum and math-methods

Course Communication

All updates and information regarding the course will be made using the announcements on ELMS – please make sure your ELMS settings do not delay announcements. I may or may not repeat in class.

Please send any questions or notifications of absences that you need to inform me preferably by email (see above).  

 

 

Campus Policies

It is our shared responsibility to know and abide by the University of Maryland’s policies that relate to all courses, which include topics like:

  • Academic integrity
  • Student and instructor conduct
  • Accessibility and accommodations
  • Attendance and excused absences
  • Grades and appeals
  • Copyright and intellectual property

Please visit www.ugst.umd.edu/courserelatedpolicies.html for the Office of Undergraduate Studies’ full list of campus-wide policies and follow up with me if you have questions.

 

Activities, Learning Assessments, & Expectations for Students

Lectures: Class time will be occupied by lectures that follow a set of notes that closely follow sections in the textbook. In a few cases I will follow a different textbook, which I will point out. I will post my notes online in this case. In addition to explaining the physical intuition behind concepts I will provide mathematical derivation of some of the more important results where the derivation is instructive. A firm grasp of quantum mechanics at approximately the advanced undergraduate level will be needed to follow parts of the lectures as well do some of the homeworks. The thorough use of quantum mechanics is what distinguishes the graduate solid state course from the undergraduate one.

Participation: The lectures assume that you are keeping track of the material of the previous lecture. This will enhance your learning and participation in the class, which is crucial to the classes success. To ensure a minimal level of participation, I will keep track of your participation through questions you ask or answer. You get full credit for participation if you ask or answer 6 questions in the semester related to the material presented in the lectures. Participation points of 2/lecture will be added to your grades within 24 hours of the lecture you participated in (i.e. asked/answered a relevant question/clarification).  I might forget to credit you for this. IT IS YOUR RESPONSIBILITY TO EMAIL ME IF I FORGET TO ADD THIS WITHIN TWO DAYS.  Late (by more than a few days) may or may not be credited depending on whether I remember.

Homework : Problem sets will be posted as assignments on ELMS. The problems can also be downloaded from the assignments folder. Homework submission should be preferably by paper in class or at instructor office. Email submission to TA is allowed but should not be hand written. In case of email submissions, please also hand in a paper submission at a later date to facilitate return at graded homeworks. Homework will be assigned roughly once a week, and is to be turned in at the beginning of class on the due date. Homework will typically be posted on Friday and due the Thursday two weeks later (i.e. about 11 days). 20% will be marked off on homework that is after the deadline. Please contact instructor to discuss in case you will not make the deadline. Homeworks turned in after the solutions are posted may not be graded. If you cannot attend class, please get your homework to me before class starts.  New assignments will be posted on the course website, along with the homework solutions.  Homework problems are carefully chosen to highlight some of the important topics covered in lecture, complete some of the important steps, as well as important applications of the ideas. It is important that you carefully complete and make sure you understand all of the homework. You are encouraged to work with others on homework, however, it is forbidden to blindly copy another person’s work. There are 7 homework sets and one will be dropped. 

 

Grades

Grades are not given, but earned.  Your grade is determined by your performance on exams, homeworks and participation in the course and is assigned based on your score according to a curve. Typically I follow a curve scheme where the median score would be graded at A-. The lowest grade would be B- (very few). The top grades will be A+ and As. The number in each category would be roughly equal depends on appropriate breaks in the score distribution. 

Of course, all this being said this rule is subject to change depending on the performance of the class. If some students do extremely poorly (e.g. score well below 40%) I might consider going below B- for the lowest grade.  On the other hand, if everyone does well (i.e. above 90%) I have no hesitation giving the entire class an A. Also, if someone scores above 80 that is a B or better independent of whether the average is above 80.

If earning a particular grade is important to you, please speak with me at the beginning of the semester so that I can offer some helpful suggestions for achieving your goal.

All assessment scores will be posted on the course ELMS page.  If you would like to review any of your grades (including the exams), or have questions about how something was scored, please email me to schedule a time for us to meet in my office.

Late work (as explained in the instruction) will not be accepted for course credit so please plan to have it submitted well before the scheduled deadline.  I am happy to discuss any of your grades with you, and if I have made a mistake I will immediately correct it.  Any formal grade disputes must be submitted in writing and within one week of receiving the grade. 

Learning Assessments

 #

Category Weight

Participation points

6

12%

Homework (out of 9 assignments)

8

88%

 

Course Schedule

 

Week 1

Monday, January 24

  • Quantum many-body physics (Ch 1)
  • Many-body Schrodinger equation (Ch 1)
  • Second quantization (Ch 2.1)

Wednesday, January 26

  • Classical and quantum fields (Ch 2.2)
  • Harmonic oscillator (Ch 2.3)
  • Phonons (Ch 2.4)

 

Week 2

Monday, January 31

  • Phonons (Ch 2.4)
  • Thermodynamics limit (Ch 2.5)

Wednesday, February 2

  • Discretization (Regularization) (Ch 2.6)
  • Quantization of number conserving fields (Ch 3.1)

Week 3

Monday, February 7

  • Review of coherent states and quanutm-classical correspondence for Harmonic oscillators

Wednesday, February 9

  • Classical limit for non-interacting Bosons (Ch 3.1.1)

Week 4

Monday, February 14

  • Noether's theorem and conserved quantities

Wednesday, February 16

  • Non-interacting Fermions (Sec 3.2, 3.3, notes)

Week 5

Monday, February 21

  • Non-interacting Fermions - basis change (3.3)
  • Deriving many-body Schrodinger equation (3.7)

Wednesday, February 23

  • Thermodynamics of non-interacting fields (3.8)
  • Jordan-Wigner transformation for spin chains(4.1)

Week 6

Monday, February 28

  • Hubbard model (Sec 4.2)
  • Anti-ferromagnetic XXZ model (Sec 4.1)

Wednesday, March 2

  • Ferromagnetic XXZ/Heisenberg model (Sec 4.1)

Week 7

Monday, March 7

  • Ferromagnetic Heisenberg model (4.1)
  • Fermi gas, pressure, classical limit (4.3.1)

Wednesday, March 9

  • Bose-Einstein Condensation (4.3.2)
  • S-matrix in the interaction representation (5.1)

Week 8

Monday, March 14

  • Driven Harmonic oscillator (5.1.1)

Wednesday, March 16

  • Wicks' theorem (5.1.2)

Week 9

Monday, March 28

  • Free fermion Green functions (5.2)

Wednesday, March 30

  • Gell-Mann-Low theorem (5.3)

Week 10

Monday, April 4

  • Spectral representation of Green function (5.3.3)

Wednesday, April 6

  • Many-body Green function/Fermion Wick's theorem (5.4)

Week 11

Monday, April 11

  • Landau-Fermi-liquid theory - quasiparticles (6.2)

Wednesday, April 13

  • Quasi-particle dispersion, interaction and entropy (6.3)

Week 12

Monday, April 18

  • Landau parameters (6.3.1)
  • Instabilities of Fermi liquids (6.4)

Wednesday, April 20

  • Collective modes in Fermi liquids (6.5)

Week 13

Monday, April 25

  • Momentum distribution, Luttinger's theorem in Fermi liquids (6.8)
  • Feynman diagrams for scattering potentials (7.1)

Wednesday, April 27

  • Linked cluster expansion (7.1,7.2.2)
  • Feynman diagram for full Green function (7.1)

Week 14

Monday, May 2

  • Feynman rules for Interaction vertex (7.1)
  • Comparing to Hartree-Fock (7.1,7.4)

Wednesday,  May 4

  • Linked-cluster with interactions - one and two particle Green functions (7.2.2)
  • Symmetry factors (7.2.1)

Week 15

Monday, May 9

  • momentum space Feynman diags (7.3)
  • Response functions (7.6)
  • Large N (7.7)

 

 

Note: This is a tentative schedule, and subject to change as necessary – monitor the course ELMS page for current deadlines.  In the unlikely event of a prolonged university closing, or an extended absence from the university, adjustments to the course schedule, deadlines, and assignments will be made based on the duration of the closing and the specific dates missed.

Course Summary:

Date Details Due