C6 Option/Advanced Quantum Theory, Michaelmas 2020
This is the webpage for the Michaelmas 2020 lecture course on "Advanced Quantum Theory". This counts as part of both the MPhys C6 Theory option, as well as the Advanced Quantum Theory course within the Oxford Masters Course in Mathematical and Theoretical Physics (MMathPhys). This course will focus on path integral techniques and many-particle quantum mechanics, and provide an introduction to Landau's theory of phases and phase transitions. It thus serves as a broad overview of the basic tools and concepts of theoretical physics.
Trinity Term Revision Class Announcement
We will have a revision session from 14.00-17.00 on Wednesday, May 19 (Week 4). The revision will cover 1-2 past papers (only the AQT topics/MT material for C6) as well as address any questions from students.
Zoom and Padlet links have been circulated and posted on the canvas/weblearn pages. Please email me if you are unable to find these.
The session will be recorded and made available via Panopto.
Organisation
Normally, this series would involve a total of around 26 hours of lectures, in Weeks 1-8. Of these, roughly hours 1-20 count for AQT; hours 21-26 will cover stochastic processes, which is an examinable part of the C6 course.
However, due to the COVID-19 crisis, we will operate in physical distanced, but electronically interactive mode. This means that rather than have hour-long lectures, you will instead be given access to a series of pre-recorded videos. Since we are liberated from the straitjacket of timetabling, the videos will not fit into neat 1-hour slots (ideas are seldom so well-behaved). Instead, they will be organised by topic in tandem with the lecture notes.
In order to allow the spontaneous and interactive exchanges that are the best part of any course (certainly for the lecturer!) these pre-recorded videos will be supplemented by:
Question & Answer sessions on Mondays at 9am, where I will be on hand "live" via Zoom to answer questions from the assigned viewing/reading of the previous week; and
"Extras and examples" ("E&Es") classes on Thursdays through the term (from 9-10 or 11 am depending on the week). These are also live Zoom sessions, in which I will go over some worked examples, and also on occasion discuss topics that are not examinable but will attempt a synthesis of different ideas in the lectures. For instance, these might include developing path integrals for many-particle Bose systems, or using the quantum-classical correspondence, Jordan-Wigner transformations, and second quantisation to solve the 2D Ising model.
These live sessions will also be recorded so that students who are unable to attend for technical or health reasons, or who are stuck in an inconvenient time zone, will be able to catch up. However, if no students turn up to attend a session, it will end at ~10 minutes past its start time, since the point is to be interactive, and no recording will be made.
Technology
All the lecture videos are accessible via Canvas/Panopto, but an introductory video about course logistics is shown here as a sample. You should all have access to these videos; contact the respective teaching offices for MMathPhys/MPhys if you are unable to view them.
For interactive sessions (Q&A and E&E), you will need to install Zoom or have a browser capable of accessing Zoom videos. You do not require a paid Zoom account for this.
Problem sessions will likely take place via Zoom or Microsoft Teams at the discretion of the individual tutor. You should be able to get Teams via your University account.
Google Jamboard will be the preferred tool for interactive live sessions. I will broadcast my Jamboard screen via Zoom, but I recommend that you also create a Google account (should you not have one already). Jamboard is particularly useful if you have a tablet, but it is also accessible via web browser. You can sketch and write equations interactively, and also flip back to earlier notes while the lecture is in session. This may also be helpful as a means of collaboration on problem sets or in discussing ideas with your classmates.
Weekly Schedule
Each week, usually on a Monday evening, I will send a list of videos that you are expected to view before the Q&A session on Monday of the subsequent week, and a list of topics and examples to be covered in the Thursday E&E session for that week.
Meeting schedules will be different in Week 1 since Oxford undergraduates will be completing Part B exams during the week.
Problem Classes
There will also be 3 problem classes this term, led by DPhil student/PDRA tutors (plus an additional one in Hilary term covering stochastic processes for thee C6 students). Each class is run in conjunction with a problem sheet, which you must complete and submit (e.g. by scanning handwritten pages or by working on a tablet) electronically to your tutors, who will then provide feedback. To guarantee that you receive feedback, the work must be completed by the deadline assigned by the respective tutor, and definitely before the problem class in order to ensure that you are prepared. Please be considerate of your tutors time as they are also juggling many challenges in the new normal. The classes usually take place in weeks 4, 6, and 8, but this may be adjusted this year. You will be assigned to a problem class in the first few weeks of term.
Lecture Plan
All videeos are available via Canvas/Weblearn
As a rough guide:
Q&A classes in week n will cover all videos assigned in week n-1.
E&E classes in week n will cover all videos assigned in week n-1, and also starred videos in week n.
Week 0
Assigned Videos: All students should view the Course Introduction video and ensure that they can access videos (e.g. via Canvas > C6 theoretical physics > Panopto Recordings or by directly looking at Panopto videos shared with you)
All students should install and configure Zoom, Teams, and (optional but recommended) Jamboard.
Week 1
Assigned Videos:
Functionals [16 min]
Gaussian Integrals [25 min]
Path Integrals and Propagators [42 min]
Free Particle Propagator [13 min]
Monday: Introductions and logistics (Students taking Part B exams can choose to watch recording later)
Thursday: No E&E class as Part B exams ongoing.
Week 2
Assigned Videos:
*Path Integrals in Quantum Statistical Mechanics [16 min]
*Finite-temperature harmonic oscillator [37 min]
Correlation Functions [18 min]
Wick's Theorem [6 min]
Probability Distribution of Position [9 min]
Feynman Diagrams [33 min]
Monday: Q&A on Week 1 Assigned Viewing
Thursday: E&E topics: Euler-Lagrange Equations, Lattice Laplacians, Matsubara sums.
Week 3
Assigned Videos:
*Quantum-Classical Correspondence [21 min]
*1D Ising Model [38 min]
*2D Ising Model [13 min]
Spontaneous Symmetry Breaking [8 min]
Monday: Q&A on Week 2 Assigned Viewing
Thursday: E&E topics: Peierls Argument; Quantum-classical correspondence between 2D Classical Ising Model and the 1+1D Quantum Ising Chain
Week 4 -- problem set 1 due
Assigned Videos:
*Independent Particles and Occupation number representation [31 min]
*Fock Space and Changes of Bases [19 min]
Building Second Quantized Operators [43 min]
Ideal Fermi Gas [53 min]
Monday: Q&A on Week 3 Assigned Viewing
Thursday: E&E topics: the Jordan-Wigner mapping and Kramers-Wannier duality for the quantum Ising Chain; Fermionic Bogoliubov transformations; exact solution of the quantum Ising chain.
Week 5
Assigned Videos:
Weakly Interacting Bose gas [62 min]
Monday: Q&A on Week 4 Assigned Viewing
Thursday: E&E topics: Path Integrals for Many-particle Bose Systems
Week 6 -- problem set 2 due
Assigned Videos:
Spin waves in Magnets [49 min]
Monday: Q&A on Week 5 Assigned Viewing
Thursday: E&E topics: Grassmann variables; Path Integrals for Many-particle Fermi Systems
Week 7
Assigned Videos:
*Phase Transitions, Critical Behaviour, and Universality [26 min]
*Landau Theory [24 min]
Fluctuations [26 min].
Gaussian Approximation [26 min].
Other Examples of Phase transitions [12 min]
Monday: Q&A on Week 6 Assigned Viewing
Thursday: E&E topics: Critical theory of the quantum Ising chain.
Week 8 -- problem set 3 due
Assigned Videos:
*Basics of Random Variables [32 min]
*Stochastic and Markov Processes
*Brownian Motion
Monday: Q&A on Week 7 Assigned Viewing -- last AQT-centric Q&A session.
Thursday: E&E topics: Brownian motion via the path integral; closing summary of AQT and C6.
Week 9
Monday: Final Q&A on Week 8 Assigned viewing (C6 only -- may be postponed to HT, or Friday of week 8)
Notes and Problems: updated Oct 30 (problem 17 & 18 swapped, problem 18 moved to PS3)
Lecture notes - path integrals and many-particle physics [permalink to live, dynamically updated file; most recent update on Nov 4 adds 20+ pages on quantum Ising chain.]
Lecture notes - Stochastic processes (C6 only) [permalink to live, dynamically updated file.]
Problem Sheet 1 [Problems 1-10, covering path integrals and transfer matrices; problem class in MT week 4]
Problem Sheet 2 [Problems 11-17, covering second quantization and applications to bosons and fermionns; problem class in MT week 6]
Problem Sheet 3 [Problems 18-21, covering spin waves and Landau theory; problem class in MT week 8.]
Problem Sheet 4 [Problems 22-26, covering stochastic processes; problem class in HT, C6 only]
Revision Problems [N.B. Question 4 on Brownian motion is outside of the AQT syllabus and need not be attempted]
Note: COVID-19 has made the standard email torrent into a deluge. Some simple guidelines to ensure you receive an answer quickly:
Please first read this page and the MMathPhys/Part C handbooks to see if your question is answered there
If your question pertains to content, then I am your point of contact whether you are enrolled in C6 and AQT
If your question pertains to organisation of this course, and you are an AQT student, I am your point of contact
If your question pertains to organisation of this course, and you are a C6 student, please contact both me and Prof. Andrei Starinets (C6 Option Coordinator)
If your question pertains to organisation of the fourth-year C6 option, please contact Prof. Andrei Starinets
If your question pertains to organisation of your year overall, examination logistics, etc., please contact Jasmine Smith (AQT) or Carrie Leonard-McIntyre (C6).
Due to the sheer volume of correspondence, emails that are misdirected or for which the resolution is clearly available in course resources may go unanswered.