How do we solve problems where the potential does not have exact solutions? This will take us to Chapter 7 of Griffiths.

22 April

• By the end of today's class, you should be able to...
• Describe how the (anti-)symmetric requirement of (femionic)bosonic wave functions results in different separations.
• Describe how the antisymmetric requirement of electrons results in the structure of the periodic table. (Chapter 5.2.2)
• Use spectroscopic notation (also Chapter 5.2.2).
• Explain why we need perturbation theory and the assumptions in our derivation.
• Materials
• Link to the day's notes

24 April

• By the end of today's class, you should be able to...
• Derive the first order corrections to wave functions and energies using the approach of time independent perturbation theory.
• Explain the concept α with regards to the strength of the electromagnetic interaction.
• Apply time-independent perturbation theory to determine the relativistic corrections to the hydrogen atom to order α⁴.
• Materials
• Link to the day's notes

Homework 8

What are the consequences of the fact that all particles of the same species are truely indistinguishable from each other? Note, I am pulling a lot from not only Chapter 5 in Griffiths, but also Chapter 18 in Baym.

15 April

• By the end of today's class, you should be able to...
• Recognize the operators which are permissible in a multi-particle system.
• Recall that fermions must have antisymmetric wave functions while bosons must have symmetric wavefunctions.
• Materials
• Link to the day's notes
• The day's slides

17 April

• By the end of today's class, you should be able to...
• (Anti)Symmetrize a wavefunction.
• Define the exchange operator.
• See how exchange forces arise from the requirement that wavefunctions be (anti)symmetric.
• Materials
• Link to the day's notes
• The day's slides

Homework 7

• Homework 7 Solutions

We had the midterm and then began to explore constructing wavefunctions for multi-particle systems.

8 April - Midterm

10 April

We began class with a debrief of the midterm, and then moved into multiple particle systems.

• By the end of today's class, you should be able to...
• Recall that for multiparticle systems, we have a single Hamiltonian and single state/wavefunction for all the particles together.
• Build a composite wavefunction for mutliple particle systems.
• In a wavefunction picture: ψ = Π_i ψ_i
• In a ket picture: | ψ > = ⊗ | ψ >_i
• Materials
• Link to the day's notes

No homework this week as there was only half a day of class!

In class last week we saw 1/2 ⊗ 1/2 = 1 ⊕ 0. What do the symbols ⊗ and ⊕ mean? How can we convert uncoupled states to coupled states without doing a lot of J_- over and over?

1 April

• By the end of today's class, you should be able to...
  • Define a tensor product.
  • Compute a tensor product.
  • Recognize that J = L + S is really J = L ⊗ 1 + S ⊗ 1 and explain why.
  • Create the coupled states of angular momentum addition using tensor products and show that the result is the same as if you used J_-.
  • Interpret the block-diagonal result in terms of ⊕
  • Read a Clebsh-Gordon Table.
• Materials: I didn't really have "notes" for this day's class, but there are several materials:
• For the discussion on tensor products, I used a lot of ChatGPT to help me do the matrix multiplication etc.
  • The summary of the ChatGPT conversation can be found here.
  • However, it made a LOT of mistakes! I am also including the full transcript of the conversation here so you can see exactly how bad it did!
• For the Clebsch-Gordon Coefficients, I want you to be able to read tables like this one in the PDG.

3 April

Today, we spent much of class discussing the recent events regarding visa revocations happening in the department.

We also discussed how folks felt about taking an exam on Tuesday given the current climate. The consensus was to go ahead.

However, we also discussed possible alternative structures to the remainder of the semester to account for the additional emotional load. We came up with a plan which switches to more of a mastery model for the rest of the semester. I am currently working to develop a new syllabus for us.

After the discussion, we did have a few minutes to discuss Bell's Inequality and how that shows that Einstein's interpretation of quantum mechanics in terms of "hidden variables" cannot possibly be correct.

• Materials
• While I will not make this part of your exam, the notes on Bell's Inequality can be found here.

No homework this week. Just prepare for your exam. However, Griffiths 4.40 may be helpful!

How to combine different angular momenta? We know how to do it classically (just add the vectors), but what about quantum mechanically?

25 March

• By the end of today's class, you should be able to...
  • Generate the first few states arising from J = 1 from the uncoupled basis of two spin-1/2 particles.
• Materials
• Link to the day's worksheet.
• The solutions to the day's quiz.

27 March

• By the end of today's class, you should be able to...
• Basically, we will debrief the worksheet.
• Add any two angular momenta quantum mechanically.
• Read the Clebsh-Gordon table.
• Materials
• Link to the day's notes.

Homework 6 - Just finish the worksheet through question 8 on page 10. As per class, you do NOT need to apply J² in problem 3.c. Instead, just write the ket |j,m>

• Solutions to the Tutorial

How to combine different angular momenta? We know how to do it classically (just add the vectors), but what about quantum mechanically?

11 March

• By the end of today's class, you should be able to...
  • Connect the two different ways of expressing angular momenta in different bases: as eigenvectors and through rotations.
  • Compute the communtation relations for J², L², S², J_i, L_i, and S_i for J = L + S.
  • Add two spin-1/2 particles.
• Materials
• Link to the day's notes.

13 March

• Today we started our tutorial on angular momentum addition.
• Materials
• Link to the day's worksheet.

Homework 5

• There is no need to do problems 4 and 5 (Griffiths 4.37 and 4.40 ) as we have not covered that material yet.
• Do add through question 1 in the tutorial (at the bottom of page 4). This is where we got on Thursday.
• Homework 5 Solutions

How does all this group theory connect to angular momentum?

4 March

• By the end of today's class, you should be able to...
• Define homomorphism and isomorphism.
• Explain why SU(2) and SO(3) are homomorphic.
• Generate representations of SU(2) for any dimensionality.
• Review angular momentum from QMI.
• Materials
• Link to the day's notes.
• Solutions to the quiz on Griffiths Section 4.4

4 March

• Today was just some QMI angular momentum review
• Materials
• Link to the day's worksheet.

Homework 4

• Homework 4 Solutions

Thinking about continuous groups of matricies, particularly in the context of time translation and rotations.

27 February

• By the end of today, you should be able to...
• 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.
• Choose a suitable basis for the angular momentum operator for a given ket basis.
• Materials
• Link to the day's notes.
• Link to the day's slides.
• Link to the quiz solutions.

27 February

• By the end of today, you should be able to...
• Show how angular momentum is the generator of rotations. In particular, the equivalence between thinking of rotations of R³ vectors and a guessed rotation operator based in Noether's Theorem of R_z(φ) = exp[-i φ L_z/ℏ].
• Classify the group of rotations on R³ vectors such as x and p as SO(3).
• Define: Lie group, generators, Lie algebra, and structure constants.
• Determine the dimensionality of a Lie group.
• Materials
• Link to the day's notes.

Homework 3

• Homework 3 Solutions

Understanding Parity as an Application of a Discrete Group.

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.
• Materials
• Link to the day's notes.
• Link to the day's slides.

19 February Discussion: A tutorial on parity.

Homework 2

• Homework 2 Solutions

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

• Homework 1 Solutions

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