1. Last hour: Matrices in QM
Use matrix algebra to solve EF/EV problems in QM (most common computational approach in QM programs).
Construct matrix representation of an operator  using ajk = <uj|Â|uk>
Resulting matrix A encodes operation of  on basis functions |uj>
Interpretation of matrix elements: Diagonal elements akk = <uk|Â|uk> is <Â> for system in state |uk> Off-diagonal elements ajk = <uj|Â|uk> describe how two states |uk> and |uk> are coupled by Â.
If the uj are EFs of the operator Â, the matrix is diagonal.
Solve det(A -1) = 0 to obtain EVs .
Plug in EVs into A (j) = j (j) to get eigenvectors, which represent the EFs of  expressed in the basis of the {uj}.

2. Learning Goals for Chapter 13 – Matrix QM
After this chapter, the related homework problems, and reading the relevant parts of the textbook, you should be able to:
construct the matrix representation of an operator for a given basis;
find the eigenfunctions and eigenvalues of an operator using matrices;
evaluate effects of matrix size on the accuracy of the results.

3. Learning Goals for Chapter 14 – The 2-State-System
After this chapter, the related homework problems, and reading the relevant parts of the textbook, you should be able to:
interpret the structure of the Hamiltonian and the contributions of its parts to the eigenvalues in a 2-state-system;
apply limiting behavior for the coupling of the levels in a 2-state-system where appropriate.