Graduate Quantum Mechanics II, Fall 2023

Basic info

Instructor: Prof. Seung-Sup Lee (이승섭), Department of Physics and Astronomy, SNU.

TA: Mr. Goojin Kwon (권구진), Department of Physics and Astronomy, SNU.

Class dates, time, and place: Tuesdays and Thursdays, 11:00–12:15, Building 56, Room 106.

Course outline: Please check the course outline (updated on Nov. 1, 2023) for the details of the course. Please check the course outline regularly, since it will be updated from time to time; critical changes will be also announced via eTL.

Main textbook: Jun John Sakurai and Jim Napolitano, Modern Quantum Mechanics (3rd Ed., Cambridge University Press, Cambridge, 2020) (ISBN: 978-1-108-47322-4). Throughout this course, the textbook will be often called “S&N”. When we refer to sections, equations, figures, problems, etc. without specifying the reference, they are assumed to be from S&N.

Student Q&A forum: In this KakaoTalk chat (closed), students can ask questions, answer them, and discuss any topic. You can join anonymously, but we recommend using your real name (a Korean name if you have one, or in the Roman alphabet if you don't). This is outside of the regular course, so feel free to Q&A in Korean!

Exam times and place: Mid-term on Oct. 26 (Th), 19:00–22:00; final on Dec. 20 (We), 19:00– 22:00; both at Building 56, Room 105 (note that it’s the room on the opposite side to the one for classes!)

Makeup classes: There will be no classes on Sep. 28 and Oct. 03 due to national holidays and on Oct. 10, Oct. 12, and Oct. 19 due to the instructor’s conference visits. Hence there will be three makeup classes over Zoom, at 19:00–20:15, on Sep. 27 (We), Oct. 04 (We), and Oct. 16 (Mo). We use the same Zoom link for all make-up classes.

Lecture materials

The list below provides the links to lecture notes, exercises, and their solutions. The lecture notes are numbered as Lxx.y (e.g., L02.1), where xx indexes a class within the semester and y indexes a topic within the class. The exercise sets are similarly numbered as Exx, and an exercise within a set as Exx.y.

Please refer to the course outline (see above) for the details: when lecture materials will be uploaded, how to prepare and submit exercise solutions, etc.

Please report if you find any mistakes or typos. Such reports will also count towards the evaluation!

[Class 26] Dec. 14 (Th)

• Q&A

[Class 25] Dec. 12 (Tu)

• [L25.1] Coherent states (lecture note)

• [L25.2] Field integral for the quantum partition function (lecture note)

[Class 24] Dec. 07 (Th)

• [L24.1] Fermi liquid theory (lecture note; typo fixed on Dec. 07)
  1. Basic statement
  2. Adiabatic continuity
  3. Renormalization of the single-particle GF

• [L24.2] Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity (lecture note)
  1. BCS Hamiltonian and mean-field approximation
  2. BCS ground state
  3. Self-consistent determination of the BCS order parameter
  4. Nambu formalism

[Class 23] Dec. 05 (Tu)

• Q&A

[Class 22] Nov. 30 (Th)

• [L22.1] Interaction picture revisited (lecture note; typos fixed on Nov. 30)

• [L22.2] Linear response theory (lecture note; typos fixed on Dec. 06)

• [L22.3] Kubo formula for the conductivity (lecture note; typos fixed on Nov. 30)

• [L22.4] Kubo formula for the dielectric function (lecture note)

• [E22] (exercise set) (solution by 김지훈)
  Deadline: Dec. 06 (We), 23:59

[Class 21] Nov. 28 (Tu)

• [L21.1] Keldysh formalism (lecture note)
  1. Three formalisms for correlation functions in quantum many-body theory
  2. Keldysh formalism (KF)
  3. Relations among retarded, advanced, and Keldysh GFs

• [L21.2] Matsubara formalism (lecture note)
  1. Matsubara GF
  2. Fourier trafo. of Matsubara GF
  3. Lehmann representation
  4. Matsubara sum
  5. Caveat to numerical analytic continuation

• [E21] (exercise set) (solution by 김지훈)
  Deadline: Dec. 04 (Mo), 23:59

[Class 20] Nov. 23 (Th)

• [L20.1] Green’s functions (GFs) (lecture note)
  1. GFs in classical electromagnetism
  2. GFs in quantum mechanics (not in many-body theory)
  3. Single-particle retarded GF of many-body systems

• [L20.2] Equation of motion, self-energy (lecture note)

• [L20.3] Lehmann representation, spectral function (lecture note)

• [E20] (exercise set) (solution by 김지훈)
  Deadline: Nov. 29 (We), 23:59

[Class 19] Nov. 21 (Tu)

• Q&A

[Class 18] Nov. 16 (Th)

• [L18.1] Symmetries (lecture note; revised on Nov. 16)
  1. U(1) symmetries
  2. SU(2) symmetries
  3. Symmetries are useful for solving Hamiltonians

• [E18] (exercise set) (solution by 김지훈)
  Deadline: Nov. 22 (We), 23:59

[Class 17] Nov. 14 (Tu)

• [L17.1] Some famous models (lecture note; updated on Nov. 14)
  1. Hubbard atom
  2. Hubbard model
  3. Anderson impurity model
  4. Heisenberg model
  5. Kondo model

• [L17.2] Schrieffer–Wolff transformation (lecture note; updated on Nov. 14)

• [E17] (exercise set) (solution by 박상현)
  Deadline: Nov. 20 (Mo), 23:59

[Class 16] Nov. 09 (Th)

• [L16.1] Second quantization (lecture note)
  1. Bosons
  2. Fermions
  3. Operators in second quantization
  4. Change of basis
  5. Field operators
  6. Example: non-interacting fermions

• [L16.2] Electromagnetic radiation in the second quantization (lecture note)

• [E16] (exercise set) (solution by 김지훈)
  Deadline: Nov. 15 (We), 23:59

[Class 15] Nov. 07 (Tu)

• [L15.1] Basics of quantum many-body physics: Intro (lecture note)

• [L15.2] Many particles in the first quantization (lecture note; updated on Nov. 09)

• [E15] (exercise set) (solution by 김지훈)
  Deadline: Nov. 13 (Mo), 23:59

[Class 14] Nov. 02 (Th)

• Review and appeal mid-term exam grading

• Q&A

[Class 13] Oct. 31 (Tu)

• Explanation of the mid-term exam problems and solutions

Mid-term exam

• Exam problems and solutions

[Class 12] Oct. 17 (Tu)

• Q&A

[Class 11] Oct. 16 (Mo) (Zoom)

• [L11.1] Hard-sphere scattering (lecture note)

• [L11.2] Low-energy scattering (lecture note)

• [L11.3] Resonance scattering (lecture note; updated on Nov. 3)

• [E11] (exercise set) (solution by 김지훈)
  Deadline: Oct. 23 (Mo), 23:59

[Class 10] Oct. 05 (Th)

• [L10.1] Optical theorem (cont'd) (lecture note)
  1. Derivation from the unitarity of the S matrix

• [L10.2] Partial-wave expansion and phase shifts (lecture note)

• [L10.3] Determination of phase shifts (lecture note)

• [E10] (exercise set) (solution by 김지훈)
  Deadline: Oct. 11 (We), 23:59

[Class 09] Oct. 04 (We) (Zoom)

• [L09.1] Born approximation (lecture note)
  1. Born approximation
  2. Example: Yukawa potential

• [L09.2] Properties of spherical harmonics (lecture note)
  1. Spherical harmonics as rotation matrices
  2. Addition theorem
  3. Spherical wave expansion

• [L09.3] Spherical waves (lecture note)
  1. Representation in the k-space
  2. Representation in the position space

• [E09] (exercise set) (solution by 김지훈)
  Deadline: Oct. 10 (Tu), 23:59

[Class 08] Sep. 27 (We) (Zoom)

• [L08.1] Scattering amplitude (lecture note)

• [L08.2] Optical theorem (lecture note)

• [E08] (exercise set) (solution by 김지훈)
  Deadline: Oct. 03 (Tu), 23:59

[Class 07] Sep. 26 (Tu)

• [L07.1] T matrix and S matrix (lecture note)
  1. Derivation by using Green's function
  2. Scattering from the future to the past

• [E07] (exercise set) (solution by 조명기)
  Deadline: Oct. 02 (Mo), 23:59

[Class 06] Sep. 21 (Th)

• [L06.1] Geometric phase (lecture note)
  1. Berry connection and curvature
  2. Correspondence to Aharonov–Bohm phase
  3. Example: Spin-1/2 in a B field

• [E06] (exercise set) (solution by 김지훈)
  Deadline: Sep. 27 (We), 23:59

[Class 05] Sep. 19 (Tu)

• [L05.1] Photoelectric effect (lecture note)

• [L05.2] Sudden approximation (lecture note)

• [L05.6] Adiabatic approximation (lecture note)
  1. Adiabatic theorem
  2. Problem with traditional adiabatic condition

• [E05] (exercise set) (solution by 김지훈)
  Deadline: Sep. 25 (Mo), 23:59

[Class 04] Sep. 14 (Th)

• [L04.1] Energy shift and decay width (lecture note; typo fixed on Sep. 14)

• [L04.2] Absorption and emission in a classical radiation field (lecture note; typo fixed on Sep. 14)
  1. Absorption and emission rates
  2. Electric dipole approximation

• [E04] (exercise set; a mistake fixed on Sep. 19) (solution by 김지훈)
  Deadline: Sep. 21 (Th), 23:59

[Class 03] Sep. 12 (Tu)

• [L03.1] Interaction picture (lecture note)
  1. Recap: Schrödinger vs. Heisenberg picture
  2. Interaction picture
  3. Example: Rabi oscillations in a two-level system

• [L03.2] Time-dependent perturbation theory (lecture note; typo fixed on Sep. 21)
  1. Dyson series in the interaction picture
  2. Constant perturbation: Fermi's golden rule
  3. Harmonic perturbation: Stimulated emission and absorption

• [E03] (exercise set) (solution by 김지훈)
  Deadline: Sep. 18 (Mo), 23:59

[Class 02] Sep. 07 (Th)

• [L02.1] Zeeman effect (lecture note; typo fixed on Sep. 7)

• [L02.2] Variational methods (lecture note)

• [E02] (exercise set; trivial typo in the set title is fixed on Sep. 11) (solution by 김지훈)
  Deadline: Sep. 13 (We), 23:59

[Class 01] Sep. 05 (Tu)

• [L01.1] Fine structure in hydrogen-like atoms (lecture note; typo fixed on Sep. 6)
  1. Relativistic correction to the kinetic energy
  2. Spin-orbit coupling
  3. Darwin term

• [E01] (exercise set) (solution by 김지훈)
  Deadline: Sep. 11 (Mo), 23:59