Quantum 2 (Physics 740)

Quantum Mechanics II meant to follow Quantum Mechanics I  (Physics 739) in the core graduate curriculum. 

This course covers some of the advanced topics of graduate quantum mechanics (like scattering and multi-particle statistics) and provides an introduction to quantum field theory for nonrelativistic systems. Quantum field theory is the language used at the frontier in almost every discipline of modern physics and is the framework for handling quantum processes that change the total number of particles.

Students are assumed to have had a first exposure to special relativity and to non-relativistic quantum mechanics and electromagnetism at a level of the core graduate courses. (Some knowledge of statistical physics is also useful, but not required.)

The main focus is on nonrelativistic systems including some many-body techniques, though electromagnetism is considered in some detail (which introduces some relativistic phenomena). Some of the consequences of merging relativity and quantum mechanics (including the Dirac equation) are discussed at the end of the term.

As the course will show, once relativity and quantum mechanics are combined it is a basic fact that all interactions necessarily change the number of particles. (This is related to the reason why antiparticles must exist.) For this reason quantum field theory seems to be the language in terms of which Nature’s fundamental laws are necessarily written. But QFT does not require relativity and this course first introduces the tools of QFT before then introducing the additional baggage required for relativistic QFT.

Although I work from my own notes (that are a work in progress, available here) sometimes it is useful to have alternative points of view. I intend to post here a selection of textbooks that can be used as alternative sources. See below for  procedural course information. The table of contents of the notes provides the most detailed syllabus, but is paraphrased here.

Although this syllabus overlaps with the syllabus of my undergraduate QFT course in this course there is relatively little overlap. This is because I adjust the topics depending on the backgrounds of students who are enrolled and the graduate level course develops them at a more advanced level (with more of an emphasis on Effective Field Theories).

SYLLABUS

  1. Review of background material
    1. Mutiparticle quantum mechanics
    2. Indistinguishable particles
    3. Occupation-number representation
  2. Creation and Annihilation Operators
    1. Bosons and Ladder Operators
    2. Fermions and Anti-commutation Relations
  3. Quantum Scattering Methods
    1. Lippmann-Schwinger Equation
    2. Time-dependent Perturbation Theory
    3. Computing operator matrix elements
  4. Simple Interactions 1
    1. Emission and Absorption
    2. Bosons and Stimulated Emission
    3. Cerenkov radiation
  5. Simple Interactions 2
    1. Fermions and Pauli Blocking
    2. Physics near a Fermi Surface
    3. Equilibrium and Detailed Balance
  6. Emergence of Classical Fields
    1. Self-energy, UV divergences and renormalization
    2. Two-particle interaction energy
    3. Coherent States and classical fields
    4. Sound waves and phonons
  7. Locality
    1. Factorization and Cluster Decomposition
    2. The Schrodinger Field and Second Quantization
    3. Interaction with an external potential
    4. Interacting Schrodinger particles
    5. Spin and Exchange Interactions
  8. Semiclassical Methods
    1. Bose-Einstein Condensation
    2. Bogoliubov transformation
    3. Feynman graphs
  9. Symmetries
    1. Symmetries in Quantum Mechanics
    2. Symmetries in QFT
    3. Spontaneous Symmetry Breaking
  10. Electromagnetic fields
    1. Classical Electromagnetism
    2. Field Quantization and Photons
    3. Casimir Energy and regularizations
    4. Atom-photon interactions
    5. Photons interacting with charged particles
    6. Electrostatic interactions
  11. Collective Effects
    1. Dielectrics
    2. Conductors and Plasmas
    3. Superconductivity
    4. Born-Oppenheimer approximation and EFTs
    5. High densities and Hartree methods
  12. Special Relativity in Quantum Mechanics
    1. Review of special relativity
    2. The Poincare group in Quantum Mechanics
    3. Which fields for which particles?
    4. Antiparticles and the spin-statistics connection
    5. C, P, T and the CPT theorem

  13. Relativistic Low-spin particles Relativistic Low-spin particles
    1. Spinless particles: the Klein-Gordon field
    2. Spinors and the Dirac Field

COURSE INFORMATION

The procedural information relevant to this course is given below.

Lectures

Classes are in person in the Winter 2024 term and their location and time will be announced here.

For best results the class should pre-read sections of the lecture notes.

Attendance to the lectures is certainly not compulsory, but if you come I do ask you to pay attention and not disrupt the class with personal conversation or social media. I will do what I can to ensure that you do not have to gnaw your own arm off to stay awake.

Textbook

There is no official course textbook; I provide notes which lay out the course much as I will present it in the lectures. The goal of these notes is to provide more detail and to provide a written record to allow you to focus on what is being said in the lectures. (I do recommend making your own notes as well, though, since writing your understanding helps absorb the material more robustly. A current version of my notes (that I continually update as the term progresses) can be found here.

For those seeking alternative viewpoints, David Tong also has a good set of lecture notes (though mostly aimed at relativistic applications) at his Cambridge University webpage.

Office Hours

Because I spend half my time at Perimeter Institute I may be hard to find in my office. So it is worth setting up any appointments in advance, and the best way to do so is to contact me by email at cburges – at – mcmaster.ca (notice the single “s” – sigh). I will also linger immediately after the scheduled lecture times since we are already together then. Otherwise, feel free to arrange another time with me on an individual basis. (I will make a point of being there for scheduled appointments, so if you do set up an appointment, please show up or give me adequate warning that you cannot make it!)

Assignments

The course work involves completing several assignments. Like any worthwhile subject, Quantum Mechanics is a contact sport and so is only really learned by doing. It is very very strongly recommended to work the assignments even if you only audit the course.

Midterm Exam

There will be a midterm test at a time and place to be announced. The midterm provides the best possible practice for the final exam, so it would be silly not to write it. Those who do not write the midterm for whatever reason can avail themselves of marking options below. 

Perhaps needless to say, you are never permitted to plagiarize material from the internet when performing a midterm exam.

Final Exam

A Final Exam will be held in the April examination session.

You are never permitted to plagiarize material from the internet when preparing any marked work, and this is especially true for final exams.

Marking Scheme

The course marks are completely based on the weekly assignments, the midterm test and the final exam. The term mark will be computed from these according to whichever of the following formulae maximizes your final mark:

  • A) Ass.: 20% Midterm: 30% Final Exam: 50%
  • B) Ass.: 20% Midterm: 0% Final Exam: 80%

You do not have to choose in advance which one you want; I will compute both and give you the maximum.

Part of the reason for providing you this menu of alternatives is to allow you to accommodate the imponderables of your own life, such as unexpected illnesses and the like. The purpose of doing so is to make this flexibility open to everybody in the class, and not just to those who wish to make special arrangements with me, or with the Associate Dean. In particular, this my preferred way to deal with MSAF applications in this class.

Additional Work and Supplemental Exam:

Additional work will NOT be available for students who might wish to improve their marks. The standard McMaster rules apply regarding the availability of supplemental exams.

Reading you your rights:

The Centre for Student Development offers free academic skill support.

Although hopefully it does not need saying, be warned that the University does not tolerate cheating, plagiarism and the like:

THE UNIVERSITY VALUES ACADEMIC INTEGRITY. THEREFORE ALL STUDENTS MUST UNDERSTAND THE MEANING AND CONSEQUENCES OF CHEATING, PLAGIARISM AND OTHER ACADEMIC OFFENCES UNDER THE CODE OF STUDENT CONDUCT AND DISCIPLINARY PROCEDURES

(see http://www.mcmaster.ca/academicintegrity for more information).

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