Understanding Parity and Expanding from our discrete groups to continuous ones, specifically time evolution.

13 February Class

• By the end of today, you should be able to...
• Expand the conceptual understanding of parity from your reading into the Z₂ group, including:
  • Creating a multiplication table.
  • Determine the groups characteristics vis-a-vis Abelian and unitarity.
  • Construct the following representations: the trivial, the 1-rep, an the 4-rep to act on 4-vectors.
• Define unitarity and explain why it is useful for physics.
• List and define other groups common in physics including U(n), SU(n), O(n), SO(n).
• Define scalar, pseudoscalar, vector, and pseudovector in terms of their parity properties.
• Determine if a Hamiltonian commutes with the parity operator.
• Describe the impacts on the wave function if the Hamiltonian commutes with parity.
• Derive the generator of the group of time translatios and show how this is connected to Noether's Theorem.
• Describe the differences between the Heisenberg and Schroedinger pictures of quantum mechanics.
• Materials
• Link to the day's notes.
• Link to the day's slides.

Homework 2

What are symmetries mathematically and physically and why are they important?

11 February Class

• By the end of today, you should be able to...
• Describe differences between simple symmetries of motion and underlying symmetries in the physical laws.
• Recognize that patterns and degeneracies are often the result of an underlying symmetry.
• Determine the symmetry underlying a pattern or degeneracy for some simple cases using calculus of variations on Lagrangians.
• State Noether's Theorem and give a few examples.
• Materials
• Link to the day's notes.
• The slides for the day.
• A cool video from Veritasium about the principle of least action

12 February Discussion: A review of time evolution from QMI

13 February

• By the end of today, you should be able to...
• List the criteria for a set to be a mathematical group.
• Define the following terms in the context of a mathematical group: closure, identity, inverse, associativity, Albelian, representation.
• Determine if a set is a group or not.
• Build a multiplication table and determine from that table if the group is Albelian or not.
• Build the multiplication table for the D3 group.
• Determine representations of groups of varying dimensionality.
• Materials
• Link to the day's notes.
• The slides for the day.

Homework 1 on QMI Review, Symmetries, and Groups

We added a lot of people so I wanted to make sure that everyone had a chance to review!

6 February

• Materials
• The worksheet reviewing QMI.
• The solutions to the worksheet.

8 February - SNOW DAY! ❄

Homework 1 on QMI Review, Symmetries, and Groups

• Homework 1 Solutions

This week, we will be introducing the course and reviewing some particularly important points from QMI

1 February

• Link to the day's notes
• The worksheet reviewing QM1