Quantum Theory II

PHYS 7230 Quantum Theory II

Instructor: Mengxing Ye (JFB 305)

Time: MW 12:55 - 2:50 PM 

Location: LCB 215

Textbooks:

• Modern Quantum Mechanics (2nd edition), J. J. Sakurai and Jim Napolitano (Cambridge University Press, 2017). ISBN 978-1108422413.

• Any edition is fine. But note that the homework problems and syllabus are based on the 2nd edition.

Course Syllabus:

Week 1-6: Review of Chapter 4 (Symmetries), Chapter 5 (Perturbation Theory), Berry phase in quantum mechanics

Week 7-14: Chapter 6 (Scattering Theory)

Week 14-15: Chapter 7 (Identical Particles), path integral in quantum mechanics

Week 16: Review

Schedule:

(Reading materials listed in parenthesis)

• Week 1: 

  • 01/06/25 -- Review: Symmetries in QM (Ch. 4)

  • 01/08/25 -- Review: Symmetries in QM: time-reversal, lattice translation; Time-independent perturbation theory: (Ch. 4, Ch. 5.1)

• Week 2: 

  • 01/13/25 -- Time-independent perturbation theory: non-degenerate case (Ch. 5.1)

  • 01/15/25 -- Time-independent perturbation theory: degenerate case (Ch. 5.2)

• Week 3: 

  • 01/22/25 -- Applications: Stark effect and hydrogen like atoms  (Stark effect: Ch. 5.1, 5.2; Hydrogenlike attoms: 5.3)

• Week 4:

  • 01/27/25 -- Applications cont'd; variational method (Hydrogenlike attoms: 5.3; variational method 5.4)

  • 01/29/25 -- Berry phase in quantum mechanics (Sakurai 3rd Edition: 5.6; Sakurai 2nd Edition: Supplement I; Additional Readings: David Tong's lecture notes on QHE: Chapter 1 & Chapter 2.1, 2.2)

• Week 5:

  • 02/03/25 -- Time-dependent potential: Interaction picture (5.5)

  • 02/05/25 -- Time-dependent perturbation 1 [Mathematica demo for the transition probability, transition rate] (5.7)

• Week 6:

  • 02/10/25 -- Time-dependent perturbation 2 (5.7, 5.9)

  • 02/12/25 -- Time-dependent perturbation 3 and application to scattering (5.9, 6.1)

• Week 7:

  • 02/19/25 -- Intro to scattering: transition rate, cross section and examples (6.1, 1st part of 6.3)

• Week 8:

  • 02/24/25 -- Scattering Amplitude 1 (6.2)

    • Additional resources 1: I have found a few online resources to learn the complex analysis that should be approachable: A lecture note: https://www.maths.ed.ac.uk/~jmf/Teaching/MT3/ComplexAnalysis.pdf. A quick note you can refer to: https://dept.rpi.edu/phys/Courses/PHYS6520/NotesOnComplexAnalysis.pdf. You can also find some lectures on youtube. The main thing to understand for quantum mechanics is how to apply the residue theorem to express an integral in terms of residues of a complex function.

    • Additional resources 2: In the lecture, I also mentioned another way to understand the Lippmann-Schwinger equation from time-independent perturbation, but one should think of the states as wave packets (to make sense of the i \eta). You can find nice discussions on it from Prof. Hitoshi Murayama's lecture notes: http://hitoshi.berkeley.edu/221B/scattering1.pdf 

  • 02/26/25 -- Scattering Amplituden 2 (6.2, 6.3)

• Week 9:

  • 03/03/25: Partial wave and phase shift 1 (6.4)

  • 03/05/25: Partial wave and phase shift 2 (6.4)

• Week 10: Spring Break

• Week 11: office hour on Friday 2-5pm

• Week 12:

  • 03/24/25: Review of Partial wave methods (6.4; for introduction of spherical Bessel function as solutions of free particle wavefunction, see Sakurai Chapter 3.7 [up to Eq. 3.7.27])

  • 03/26/25: Eikonal approximation (high energy state scattering) (6.5)

• Week 13:

  • 03/31/25: Low energy scattering and bound state (6.6; Mathematica Notebook; A very nice discussions on this topic: Hitoshi Murayama's lecture note on scattering)

  • 04/02/25: Resonance (6.7; A very nice discussions on this topic: Hitoshi Murayama's lecture note on scattering)

• Week 14:

  • 04/07/25: Resonance cont'd and symmetries in scattering 

  • 04/09/25: Permutation and 2 identical particles (7.1-7.3)

• Week 15:

  • 04/14/25: Helium and multiparticle states (7.4-7.5)

  • 04/16/25: Path integral in quantum mechanics 

• Week 16:

  • 04/21/25: Overview

• Week 17:

  • 04/28/25: Final take home exam