Graduate Quantum Mechanics I, Spring 2023

Basic info

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

TA: Mr. Sanghyun Park (박상현), 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 Apr. 20, 2024) 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, 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!

Videos and lecture notes from the course in 2023-1: For your information, we provide the recorded videos and the lecture notes used for the previous Graduate Quantum Mechanics I (2023-1) course at the bottom of this page. The previous course was in a flipped-classroom format, so the videos were prepared to be as self-contained as possible. Note, however, that the scope, notations, etc. for the previous course can differ from those for the current course.

Exam times and places: Mid-term on April 22 (Mo), 19:00–22:00; final on June 18 (Tu), 19:00–22:00; both at Building 28, Room 101.

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!

[Mid-term exam] Apr. 22 (Mo)

• Exam problems and solutions

[Class 13] Apr. 18 (Th)

• Q&A session only. No exercises!

[Class 12] Apr. 16 (Tu)

• [L12.1] SU(2) and SO(3) (lecture note)
  1. Definition of a group, SU(2), and SO(3); 2. Generators of SU(2) and SO(3); 3. SU(2)–SO(3) homomorphism

• [L12.2] Angular momentum (lecture note)
  1. Angular momentum operators as generators of rotation; 2. Transformation of states and operators according to a rotation; 3. Example: spin-1/2

• [E12] (exercise set)
  Deadline: Apr. 22 (Mo), 23:59

[Class 11] Apr. 11 (Th)

• [L11.1] Gauge transformations: scalar potentials (lecture note)
  1. Time-dependent potential is trivial if uniform; 2. Example: gravity

• [L11.2] Gauge transformations: vector potential (lecture note)
  1. Recap: electromagnetism and scalar/vector potentials; 2. Hamiltonian of a charged particle in an EM field; 3. Equation of motion; 4. Schrödinger wave equation and probability current; 5. Aharonov–Bohm effect

• [E11] (exercise set) (solution by 박창현)
  Deadline: Apr. 17 (We), 23:59

[Class 10] Apr. 09 (Tu)

• [L10.1] Propagators (lecture note; revised on Apr. 09)
  1. Properties of propagators; 2. Example: free particle

• [L10.2] Feynman's path integrals (lecture note; revised on Apr. 09)
  1. Feynman's formulation; 2. Equivalence between Feynman's path integral and the Schrödinger wave equation

• [E10] (exercise set) (solution by 유지민 & 이성민)
  Deadline: Apr. 15 (Mo), 23:59

[Class 09] Apr. 04 (Th)

• [L09.1] Linear potential (lecture note; revised on Apr. 04)
  1. Solution in terms of the Airy function; 2. Physical example: "bouncing ball"

• [L09.2] Wentzel–Kramers–Brillouin (WKB) (semiclassical) approximation (lecture note; revised on Apr. 04)
  1. Solution away from the classical turning points; 2. Solution near the classical turning points; 3. Matching conditions; 4. "Slowness" criterion; 5. Example: "bouncing ball" revisited

• [E09] (exercise set) (solution by 김현우 & 박창현 & 정기복)
  Deadline: Apr. 10 (We), 23:59

[Class 08] Apr. 02 (Tu)

• [L08.1] Schrödinger’s wave equation (lecture note)
  1. Time-dependent wave equation; 2. Time-independent wave equation; 3. Example: Free particle with periodic boundary conditions

• [L08.2] Simple harmonic oscillator revisited (lecture note)
  1. Derivation of the Hermite polynomials and their generating function

• [E08] (exercise set) (solution by 유지민 & 정기복)
  Deadline: Apr. 08 (Mo), 23:59

[Class 07] Mar. 28 (Th)

• [L07.1] Simple harmonic oscillator (lecture note)
  1. Ladder operator method; 2. Energy eigenstates in the position space; 3. Time evolution

• [L07.2] Coherent states (lecture note)

• [E07] (exercise set) (solution by 정기복)
  Deadline: Apr. 03 (We), 23:59

[Class 06] Mar. 26 (Tu)

• [L06.1] Time evolution: Heisenberg picture (lecture note; revised on Mar. 26)
  1. Heisenberg equation of motion; 2. Free particles and Ehrenfest theorem

• [L06.2] Energy-time uncertainty relation (lecture note)

• [E06] (exercise set) (solution by 정기복)
  Deadline: Apr. 01 (Mo), 23:59

[Class 05] Mar. 21 (Th)

• [L05.1] Time evolution: Schrödinger picture (lecture note)
  1. Time evolution operator; 2. Schrödinger equation for the time-evolution operator; 3. Time evolution of density operator

• [L05.2] Non-Hermitian Hamiltonians (lecture note)

• [E05] (exercise set; typo fixed on Mar. 26) (solution by 이성빈)
  Deadline: Mar. 27 (We), 23:59

[Class 04] Mar. 19 (Tu)

• [L04.1] Direct sum, tensor product, partial trace (lecture note)
  1. Direct sum; 2. Tensor product; 3. Partial trace

• [L04.2] Singular value decomposition and Schmidt decomposition (lecture note)

• [L04.3] Entanglement and Bell inequalities (lecture note)
  1. Bipartite entanglement, entanglement entropy; 2. CHSH inequality

• [E04] (exercise set) (solution by 유지민)
  Deadline: Mar. 25 (Mo), 23:59

[Class 03] Mar. 12 (Tu)

• [L03.1] Baker–Hausdorff lemma and Baker–Campbell–Hausdorff formula (lecture note; typos fixed on Mar. 12)

• [L03.2] Density operator (lecture note)
  1. Density operator; 2. Quantum statistical mechanics

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

[Class 02] Mar. 07 (Th)

• [L02.1] Unitary transformation (lecture note)
  1. Change of basis; 2. Diagonalization; 3. Analytic functions of operators; 4. Unitary as the exponential of a Hermitian; 5. Trace

• [L02.2] Position and momentum (lecture note)
  1. Translation operator; 2. Momentum operator as a generator of translation

• [E02] (exercise set) (solution by 정기복)
  Deadline: Mar. 13 (We), 23:59

[Class 01] Mar. 05 (Tu)

• [L01.1] Dirac's bra-ket notation (lecture note)
  1. Kets, bras, and operators in the Hilbert space; 2. Matrix representations; 3. Example: spin-1/2

• [L01.2] Measurements and uncertainty relations (lecture note; minor revision on Mar. 04)
  1. Measurements, compatible (commuting) observables; 2. Uncertainty relations: Robertson–Schrödinger relation and its proof

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

Videos and lecture notes from 2023-1

Note: For mini-lectures given during classes and supplemental materials, only the lecture notes are available. Also, the lecture notes might have been updated after recording the videos, so the notes shown in the videos might be different from those uploaded here.

• [L01.1] Dirac's bra-ket notation (lecture note) (video)
  1. Kets, bras, and operators in the Hilbert space; 2. Matrix representations; 3. Answers to questions during the class

• [L01.2] Measurements and uncertainty relations (lecture note) (video)
  1. Measurements, compatible (commuting) observables; 2. Uncertainty relations (Robertson, Schrödinger)

• [L02.1] Unitary transformation (lecture note) (video)
  1. Change of basis; 2. Diagonalization; 3. Unitary as matrix exponential; 4. Trace

• [L02.2] Position and momentum (lecture note) (video)
  1. Translation operator; 2. Momentum operator as a generator of translation
  (In the video, my explanation was unclear regarding the commutation of the position and translation operators. The note is revised to make it clearer; the changes are highlighted by light green underlines.)

• [L02.4] Baker–Hausdorff lemma and Baker–Campbell–Hausdorff formula (lecture note)

• [L03.1] Density operator (lecture note) (video)
  1. Density operator; 2. Quantum statistical mechanics

• [L04.1] Direct sum, tensor product, partial trace (lecture note) (video)
  1. Direct sum; 2. Tensor product; 3. Partial trace

• [L04.2] Entanglement and Bell inequalities (lecture note) (video)
  1. Bipartite entanglement; 2. CHSH inequality

• [L05.1] Time evolution: Schrödinger picture (lecture note) (video)
  1. Time evolution operator; 2. Schrödinger equation for the time-evolution operator; 3. Time evolution of density operator; 4. Energy-time uncertainty relation

• [L06.1] Time evolution: Heisenberg picture (lecture note) (video)
  1. Heisenberg equation of motion; 2. Example: free particles

• [L07.1] Simple harmonic oscillator (lecture note) (video)
  1. Ladder operator method; 2. Energy eigenstates in the position space; 3. Time evolution

• [L08.1] Schrödinger’s wave equation (lecture note) (video)
  1. Time-dependent wave equation; 2. Time-independent wave equation; 3. Example: Free particle with periodic boundary conditions

• [L08.2] Simple harmonic oscillator revisited (lecture note) (video)
  1. Derivation of the Hermite polynomials and their generating function

• [L09.1] Linear potential (lecture note) (video)
  1. Solution in terms of the Airy function; 2. Physical example: "bouncing ball"

• [L09.2] Wentzel–Kramers–Brillouin (WKB) (semiclassical) approximation (lecture note) (video)
  1. Solution away from the classical turning points; 2. Solution near the classical turning points; 3. Matching conditions; 4. "Slowness" criterion; 5. Example: "bouncing ball" revisited

• [L10.1] Propagators (lecture note) (video)
  1. Properties of propagators; 2. Example: free particle

• [L10.2] Feynman's path integrals (lecture note) (video)
  1. Composition property of propagators; 2. Equivalence of the Feynman's path integral to the Schrödinger wave equation

• [L11.1] Gauge transformations: scalar potentials (lecture note) (video)
  1. Time-dependent potential is trivial if uniform; 2. Example: gravity

• [L11.2] Gauge transformations: vector potential (lecture note) (video)
  1. Recap: electromagnetism and scalar/vector potentials; 2. Hamiltonian of a charged particle in an EM field; 3. Equation of motion; 4. Schrödinger wave equation and probability current; 5. Aharonov–Bohm effect

• Mid-term exam (problems)

• [L13.1] Angular momentum (lecture note) (video)
  1. Angular momentum operators as generators of rotation; 2. Transformation of states and operators according to a rotation; 3. Example: spin-1/2

• [L14.1] SU(2) and SO(3) (lecture note) (video)
  1. Definition of a group, SU(2), and SO(3); 2. Generators of SU(2) and SO(3); 3. SU(2)–SO(3) homomorphism