Course Information

Lectures: MWF at 1:10pm in our Zoom Meeting Room (Main Lecture).

Textbook: A Brief Introduction to Physics for Mathematicians by I. Dolgachev

Syllabus: The course syllabus contains the basic guidelines for the course.

Office Hours: Wednesdays 3-4pm at the corresponding Zoom Meeting Room. Always feel free to ask after class.

Important Dates: First Day (Sep 30), Midterm Test (Oct 30), Final Exam (Week of Dec 14).

Problem Sets: Weekly assignments are due on Fridays at the beginning of class, to be submitted through Gradescope.

Problem Sets

1. Problem Set 1: Due Friday Oct 9 and available Wed Sep 30. (TeX File PSet1).
   
   
2. Problem Set 2: Due Friday Oct 16 and available Thu Oct 8. (TeX File PSet2).
   
   
3. Problem Set 3: Due Friday Oct 30 and available Fri Oct 16. (TeX File PSet3).
   
   
4. Problem Set 4: Due Friday Nov 13 and available Sun Nov 1. (TeX File PSet4).
   
   
5. Problem Set 5: Due Friday Dec 11 and available Mon Nov 23. (TeX File PSet5).
   
   

Math Diaries

1. Wednesday Sep 30: Introduction to MAT-265.
   
   Aspects of Classical and Quantum Mechanics.
   Lagrangian and Hamiltonian Formalisms.
   
   
2. Notes for Lecture 1.
   
   
3. Friday Oct 2: Lagrangian Formalism.
   
   Critical Points of the Action Functional. Euler-Lagrangian Equations.
   Recovering Newton's Equations. Generalizations and Example of Geodesics.
   
   Textbook Reading (Oct 3): First Part of Lecture 1.
   
4. Notes for Lecture 2.
   
   
5. Monday Oct 5: Hamiltonian Formalism.
   
   Dynamics in Hamiltonian Formalism. The symplectic geometry of the cotangent bundle.
   Properties of the symplectic form and Hamilton's equations.
   
   Textbook Reading (Oct 5): Second Part of Lecture 1.
   
6. Notes for Lecture 3.
   
   
7. Wednesday Oct 7: Noether's Symmetry Principle.
   
   Infinitesimal Symmetries and Constants of Motion.
   The Statement of Noether's Theorem.
   Conservation of Energy. Central Potentials.
   
   Textbook Reading (Oct 7): First Part of Lecture 2.
   Notes for Lecture 4.
   
   
   
8. Friday Oct 9: Symmetries in Hamiltonian Dynamics.
   
   Constants of Motions in Hamiltonian Formalism.
   Poisson Bracket and Symmetry Condition.
   Example of Radial Potential (Kepler's Problem).
   
   Textbook Reading (Oct 9): Second Part of Lecture 2.
   Notes for Lecture 5.
   
   
   
9. Monday Oct 12: Poisson Dynamics.
   
   Poisson Bracket: Hamilton's Equations and Constants of Motion.
   Hamiltonian Vector Fields. Poisson Bracket in the Cotangent bundle.
   Dynamics in Poisson manifolds. Example of a Rigid Body (Part I).
   
   Textbook Reading (Oct 12): First Part of Lecture 3.
   Notes for Lecture 6.
   
   
   
10. Wednesday Oct 14: Rigid Body Dynamics.
    
    Matrix Poisson Bracket and Cross Products in 3-space.
    Infinitesimal Rotations. Rigid body: Kinetic Energy and Hamiltonian.
    Equations of Motions. Constants of Motion and Angular Momentum.
    
    Textbook Reading (Oct 14): Second Part of Lecture 3.
    Notes for Lecture 7.
    
    
    
11. Friday Oct 16: Integrable Systems.
    
    Summary of Classical Dynamical Systems. Integrable Systems.
    The Arnol'd-Liouville Theorem: Lagrangian Torus Fibrations.
    Example of the Spherical Pendulum. (Infinite Dimensional Examples.)
    
    Textbook Reading (Oct 16): Last Part of Lecture 3.
    Notes for Lecture 8.
    
    
    
12. Monday Oct 19: Probability in Physics.
    
    Quantum Mechanics and Different Quantizations. Examples.
    Observables and States. Expectation values and density functions.
    Heisenberg's Picture and Schrodinger's Picture. Equivalency.
    
    Textbook Reading (Oct 19): First Part of Lecture 4.
    Notes for Lecture 9.
    
    
    
13. Wednesday Oct 21: The Heisenberg Uncertainty Principle.
    
    Observables and States. The case of Hermitian operators in finite dimensions.
    Hermitian operators as observables: Poisson bracket and Jordan product.
    The Heisenberg Uncertainty Principle and its Proof. Pure states and projections.
    
    Textbook Reading (Oct 21): Lecture 4.
    Notes for Lecture 10.
    
    
    
14. Friday Oct 23: Matrix Quantum Mechanics.
    
    Old Quantum Theory: Bohr–Sommerfeld quantization condition.
    Heisenberg's Matrix Formulation. Matrices in the Quantum Harmonic Oscillator.
    The evolution of Hermitian matrices observables. Eigenvectors as Pure states.
    
    Textbook Reading (Oct 23): The theoretical background is Lecture 4.
    Notes for Lecture 11.
    
    
    
15. Monday Oct 26: Schrödinger's Equation.
    
    Solution to the quantum equations of motions.
    The quantum analogue of the Hamiltonian flow.
    Evolution of observables and states. Equivalence.
    Evolution of pure states: Schrödinger's equation.
    
    
    Textbook Reading (Oct 26): Lecture 5.
    Notes for Lecture 12.
    
    
    
16. Wednesday Oct 28: Quantum Particle on a Circle.
    
    Motivation for problem and Chemistry Example. Setup.
    Energy operator. Stationary states and Possible Energies.
    Possible momenta: expected values and deviations.
    Position value and deviation. Mixed states.
    
    Textbook Reading (Oct 28): Combination of Lectures 5 and 6.
    Notes for Lecture 13.
    
    
    
17. Friday Oct 30: The Stone-Von Neumann Theorem.
    
    Motivation for problem and Chemistry Example. Setup.
    Energy operator. Stationary states and Possible Energies.
    Possible momenta: expected values and deviations.
    Position value and deviation. Mixed states.
    
    Textbook Reading (Oct 30): Combination of Lectures 6 and 9.
    Notes for Lecture 14. Also Quantum Theory for Mathematicians (Chapter 14).
    See Chapter 1.5 and Chapter 4.4 of G. Folland's book Harmonic Analysis in Phase Space,
    where Stone Von-Neumman and Groenewold-van Hove's Theorems are proven.
    
    
    
18. Monday Nov 2: Weyl's Calculus and Integral Operators.
    
    The canonical quantization of an observable. Kernel Description.
    Properties and examples of quantized observables. Action on Wave Packets.
    Symbol calculus and Moyal Product. Calderon-Vaillancourt's Theorem.
    Symplectomorphisms and Quantization. Microsupport and Propagation.
    
    Textbook Reading (Nov 2): Lecture 6 and Harmonic Analysis in Phase Space (Chapter 2).
    Notes for Lecture 15. See also T. Tao's entry on Weyl Quantization.
    
    
    
19. Wednesday Nov 4: The Quantum Harmonic Oscillator I.
    
    Canonical quantization of Hamiltonian. Factorization of operators.
    Ladder Method. The extended Heisenberg Lie algebra and its representations.
    The Fundamental Computation. Possible Energies and Stationary States.
    
    Textbook Reading (Nov 4): Lecture 7.
    Notes for Lecture 16 (Part I).
    
    
    
20. Friday Nov 6: The Quantum Harmonic Oscillator II.
    
    Summary of the Eigenstates for the Hamiltonian Operator.
    Expected values and standard deviations of position and momenta.
    Expected Potential and Kinetic Energies for the Stationary States.
    Expected position in a superposition state. Expected momentum and time evolution.
    Relation between classical and quantum. Recovering the Heisenberg picture.
    
    Textbook Reading (Nov 6): Lecture 7.
    Notes for Lecture 16 (Part II).
    
    
    
21. Monday Nov 9: Central Potentials.
    
    Structure of Stationary states for a Hamiltonian with radial potential energy.
    Rotational Symmetries of Central Systems and decomposition of representations.
    Irreducible Representations of SO(3): Harmonic Polynomials and Fundamental Computation.
    Spherical Harmonics and eigenvectors for the irreducible representations of SO(3).
    
    Textbook Reading (Nov 9): Lecture 8.
    Notes for Lecture 17.
    
    
22. Friday Nov 13: Free particle in 2-sphere and Hydrogen Atom.
    
    Central Potential: SO(3) Irreps and 1D Schrödinger Problem.
    Free Particle in the 2-sphere: Spherical Harmonics and Vibrations.
    Hydrogen Atom: stationary states and their form. Representations of SU(3).
    
    Textbook Reading (Nov 13): Lecture 8.
    Notes for Lecture 18.
    
    
23. Monday Nov 16: Coherent States and Propagators.
    (Special Event: guest lectures by A. Waldron)
    
    Coherent States. Laser Example.
    Resolution of Unity. Propagators.
    Notes for Lecture 19.
    
    
24. Wednesday Nov 18: Quantum Propagators and Path Integrals.
    (Special Event: guest lectures by A. Waldron)
    
    Path Integral. Feynamn-Dirac Interpretation.
    Propagator as an integral: resolutions of unity trick.
    Discretization of Time and Continuous Limit: The Path Integral.
    
    Notes for Lecture 20.
    
    
25. Friday Nov 20: Path Integral and Feynman Diagrams.
    (Special Event: guest lectures by A. Waldron)
    
    Semi-classical Approximation. Path Integral Quantization.
    Euler-Lagrange. Propagator Approximation. Graphical Expansions.
    Feynman Diagrams. Feynman Rules and Graph Series.
    
    Notes for Lecture 21.
    
    
26. Monday Nov 23: Classical Field Theory: Klein-Gordon
    
    Field Theory. The Klein-Gordon Lagrangian and its Equation of Motion.
    Solutions to Klein-Gordon Equation via Fourier. Wave superposition.
    Hamiltonian Formulation. Observable in Field Theory. Hint at Quantizing.
    
    Notes for Lecture 22.
    
    
27. Wednesday Nov 25: Quantization of Klein-Gordon Field Theory
    
    Canonical Quantization of the Klein-Gordon Field Theory.
    Vacuum Wavefunction(al). Particles as excitations of a quantum field.
    Fock Space, normal ordering and complex Klein-Gordon Field. Dirac Field.
    Path Integral Quantization of Klein-Gordon. Computation of Partition Function.
    
    Notes for Lecture 23.
    Informal OH Notes of Nov 25.
    
    
28. Monday Nov 30: Yang-Mills Fields.
    Notes for Lecture 24.
    
    
29. Wednesday Dec 2: Connections, gauge and Curvature.
    Notes for Lecture 25.
    
    
30. Friday Dec 4: Donaldson Theory.
    Notes for Lecture 26.
    
    
31. Monday Dec 7: Path Integral Quantization in Gauge Theory.
    Notes for Lecture 27.
    
    

Further Resources

Books on Classical Mechanics: My favourite book on this topic is "Mathematical Methods of Classical Mechanics" by V.I. Arnol'd. The book "Physics for Mathematicians, Mechanics I" by M. Spivak is rather comprehensive, and "Mathematical Physics: Classical Mechanics" by A. Knauf can work well for advanced topics. From the perspective of physics, the classical books on the subject are "Classical Mechanics" by H. Goldstein and "Mechanics: Volume 1" by L.D. Landau and E.M. Lifschitz.

Books on Quantum Mechanics: The material in the main textbook and the lectures can be complemented with the notes "Quantum Mechanics for Mathematicians" by L.A. Takhtajan, and either of the books "From Classical to Quantum Mechanics" by G. Esposito, "Quantum Theory for Mathematicians" by B.C. Hall or "Principles of Quantum Mechanics" by R. Shankar.

Standard sources in the topic, more focused on operator theory, include "Modern Quantum Mechanics" by J.J. Sakurai and "Mathematical Methodsin Quantum Mechanics" by G. Teschl. See also "Introduction to Quantum Mechanics" by D.J. Griffiths and D.F. Schroeter, and "A Student's Guide to the Schrödinger Equation" by D.A. Fleisch. Finally, the book "Quantum Theory, Groups and Representations: An Introduction" by P. Woit is a fantastic introduction to quantum mechanics from the perspective of Lie groups and their representations.

Lighter Reading on Quantum Mechanics: Both "Basic concepts of quantum mechanics" by L.V. Tarasov and "Quantum Mechanics: The Theoretical Minimum" by L. Susskind and A. Friedman are gentle introductions to the subject. One might also benefit from "The Feynman Lectures on Physics. Volume III: Quantum Mechanics" by R. Feynman.

Video resources: MIT's OpenCourseWare is a good source of Video Lectures and Lecture Notes. Among others, they include "Quantum Physics I", "Quantum Physics II", "Quantum Physics III" and "Introductory Quantum Mechanics I". L. Susskind's lighter series "Online Lectures on Modern Physics" at Stanford's Continuing Studies Series can be good to watch for general background. Finally, the IIT Madras has the video lectures "Quantum Mechanics". There are many additional resources online, let me know if you are interested in more references for a particular topic.