Physics 212 — Fundamentals of Quantum Mechanics II

Objectives and Goals

This is the second of the two course sequence designed for senior undergraduate and beginning graduate students whose aim is to develop a strong foundation in the formalism and application of quantum mechanics. This is done using the examples of a variety of idealized physical systems in which the student is expected to learn how to identify the relevant degrees of freedom, formulate the questions relevant to understanding the quantum dynamics that can be answered, and learn various analysis techniques that can help find answers to the questions posed.

The course provides exposure to the following topics:

  • Spin and Two State Systems
  • Angular Momentum Addition
  • Time Independent Perturbation Theory
  • The real Hydrogen Atom
  • Time dependent Perturbation Theory
  • Interaction of Light with Atoms
  • The Semiclassical Approximation
  • Tunneling in Quantum Mechanics
  • The Variational Approximation
  • The Adiabatic Approximation
  • Basics of Scattering Theory
  • Periodic Potentials and Crystals
  • Many Body Quantum Mechanics
  • Identical Particles: Fermions versus Bosons

Methods and Approach

Prerequisites:
Phy 211 or successful performance on a placement exam; Mth 104, Mth 111, Phy 142L or equivalent.
Format:
Lectures generally involve blackboard presentations by the professor, but student participation is encouraged. Problem sets generally consist of 7-8 problems designed to take approximately 10 hours of concentrated effort to complete. Students are encouraged to discuss the problems with their peers, but are required to write solutions independently. Students are responsible for all material covered in the lectures and in the problem sets. As with any course in physics, it is impossible to overemphasize the importance of working problems.
Texts:
The primary text for the course is typically Cohen-Tannoudji, Diu, and Laloe Vol. II or Shankar. Students are encouraged to consult additional texts such as:
  • Liboff, Introductory Quantum Mechanics.
  • J.J. Sakurai, Modern Quantum Mechanics.
  • D.J. Griffiths, Introduction to Quantum Mechanics
  • Eugene Merzbacher, Quantum Mechanics
Exams and Grades:
There is a final exam for the course and two in-class exams. Exams are designed to test each student's grasp of the fundamental concepts. Grades for the course are determined by homework (35%), two midterm (15% each), and the final exam (35%).

Sample Syllabus

(Each bullet represents 75min class.)

  • Lecture 1: Spin: how it arises in Nature and its connection with angular momentum and rotations
  • Lecture 2: Hilbert Space of Spin: Introduction to Direct Product Spaces
  • Lecture 3: Kinematics of Spin, Charged particle with Spin in a Magnetic Field, the Pauli Hamiltonian
  • Lecture 4: Dynamics of Spin in a Precessing Magnetic Field
  • Lecture 5: Combining orbital and spin degrees of freedom, angular momentum addition, two spin half particles
  • Lecture 6: The General Problem of angular Momentum Addition
  • Lecture 7: Time independent perturbation theory, application to anharmonic oscillator and other simple problems
  • Lecture 8: Degenerate perturbation theory: Simple Applications
  • Lecture 9: First Mid Term Exam
  • Lecture 10: Fine structure of the Hydrogen Atom
  • Lecture 11: Nuclear size effect, Hyperfine Interactions
  • Lecture 12: Time Dependent perturbation Theory, First Order Result, simple applications
  • Lecture 13: Interaction of an Atom with an EM Field, Fermi's Golden Rule
  • Lecture 14: Higher Orders and the Dyson Series
  • Lecture 15: Fine structure of the Hydrogen Atom
  • Lecture 16: Adiabatic Approximation, Berry's Phase, etc..,
  • Lecture 17: Variational Approximation, simple applications
  • Lecture 18: WKB Approximation, One dimensional Applications, connection formulas
  • Lecture 19: Second Midterm Exam
  • Lecture 20: Application to Tunneling, Double Well Problem, Ammonia Molecule
  • Lecture 21: Basics of Scattering Theory, One dimensional Application
  • Lecture 22: Central Potential Scattering, Scattering amplitude, Born Approximation
  • Lecture 23: scattering phase shifts and partial wave expansion
  • Lecture 24: Multi-particle systems, Identical Particles, Bosons versus Fermions
  • Lecture 25: Multi-Electron Systems, The Helium Atom, Periodic Table
  • Lecture 26: Band Theory of electrons, Bloch functions
  • Lecture 27: Introduction of temperature in Quantum Mechanics, pure versus impure state, density matrix
  • Lecture 28: Black Body radiation and a free electron gas
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