Course description:
This course provides an advanced-level introduction of quantum mechanics for students in chemistry, materials science, and related disciplines. Building on the foundations established in undergraduate physical chemistry, this course develops a deeper understanding of the mathematical structure and physical principles underlying quantum mechanics.
Rather than viewing quantum mechanics simply as yet another theory for describing the microscopic world, we will explore it as a fundamentally new framework for understanding nature. Through the concepts of Hilbert space, symmetry, and representation theory, we will emphasize how abstract mathematical formalism leads to powerful physical intuition and reveals principles that have no classical counterpart. Although its abstract formalism can appear challenging at first, this course guides students through the underlying mathematical structure to develop new physical intuition. The course also highlights connections between quantum mechanics and modern chemistry, illustrating how this formalism underlies our understanding of molecular structure, spectroscopy, many-electron systems, and contemporary quantum materials.
Prerequisites: At least one-semester of quantum mechanics at the level of Griffiths. Basic mathematics including linear algebra and multivariable calculus.
Course website and lecture notes: https://sites.google.com/bc.edu/chem6676
Topics:
1. Mathematical formalism (linear vector spaces, linear operators, representation theory, translational symmetry)
2. Quantum dynamics (Time evolution operator, Schrodinger equation, quantum mechanics postulates, Schrodinger/Heisenberg pictures)
3. Model quantum systems (Harmonic oscillators, central force problems)
4. Theory of angular momentum (rotation in 3D, rotation in quantum mechanics, representation of angular momentum operator and rotation, OAM, spin, coupling of angular momentum, hydrogen atom)
5. Approximation methods (stationary perturbation theory, variational methods)
6. Multielectron systems (identical particles, symmetrization postulate, Hartree-Fock equations, electronic structure of molecules).
Primary references:
J. J. Sakurai & Napolitano, Modern Quantum Mechanics
A. Messiah, Quantum Mechanics
Atkins & Friedman, Molecular Quantum Mechanics
There is no required textbook for this course. Comprehensive lecture notes will be provided and will serve as the primary course material. The structure of this course is developed based on Sakurai’s Modern Quantum Mechanics. Messiah’s book is used extensively for its elegant and rigorous development of the mathematical formalism. While both Sakurai’s and Messiah’s books are standard textbooks for physics majors, Atkins & Friedman’s book provides valuable chemical perspectives and applications that complement the theoretical framework presented in this course.
Assessment (tentative):
Homework, 6 problem sets (30%). Problem sets are due at noon on the due date. Late submissions will receive a maximum of 50% credit.
Midterm exam, closed book, in-class, 1h (20%). Covering Topics 1-3, and representation of AM in Topic 4
Final exam, closed book, 3h, December 16, 9:00 a.m. (50%)
Lecture notes: (These notes are updated continuously; please use the online version rather than downloading local copies)
Lecture 1 – Mathematical Formalism (link).
Problem sets:
PS #1 – due 9/11 (link).