This document includes two main problems related to linear algebra and quantum mechanics.

Problem 1: Matrix Calculations

1. Problem Statement

   • Given a $2 \times 2$ matrix $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, you need to:

     • (a) Calculate $\ln A$ using the diagonalization method.
     • (b) Compute $\exp(i A t)$ using the series expansion method.

   • For the series expansion, the following Taylor series expansions are provided: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}, \quad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}, \quad \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}.$

2. Solution Outline

   • (a) For $\ln A$: Diagonalize the matrix $A$, find the eigenvalues and eigenvectors, and express $\ln A$ in terms of these components.
   • (b) For $\exp(i A t)$: Use the series expansion for the matrix exponential, incorporating the identity and the matrix $A$ terms iteratively.

Problem 2: Quantum Mechanics in Hilbert Space $\mathcal{H}_3$

1. Problem Statement

   • Given an orthonormal basis $\{|e_1\rangle, |e_2\rangle, |e_3\rangle\}$, and a quantum state: $|\psi\rangle = \frac{i}{2} |e_1\rangle + \sqrt{\frac{3}{8}} \left( |e_2\rangle - |e_3\rangle \right),$ answer the following:
     • (a) Show that $|\psi\rangle$ is normalized.
     • (b) Consider an observable $\hat{O}$ represented by the matrix $\begin{pmatrix} -1 & 0 & 0 \\ 0 & 2 & i \\ 0 & -i & 2 \end{pmatrix}$. Find possible measurement outcomes for this observable.
     • (c) Calculate the probability of measuring outcome 3 for the observable $\hat{O}$.
     • (d) Determine the state of the system immediately after a measurement records outcome 3.

2. Solution Outline

   • (a) Verify normalization by calculating $\langle \psi | \psi \rangle = 1$.
   • (b) Diagonalize $\hat{O}$ to find eigenvalues, which represent the possible measurement outcomes.
   • (c) Use the projection of $|\psi\rangle$ onto the eigenvector associated with outcome 3 to compute the probability.
   • (d) After obtaining outcome 3, project $|\psi\rangle$ onto the corresponding eigenvector and normalize.

Would you like me to work through any of these parts in detail?

Follow-up Questions:

1. What are the eigenvalues and eigenvectors of matrix $A$?
2. How does the series expansion method apply to non-commuting matrices?
3. Why is diagonalization useful in calculating $\ln A$?
4. What physical meaning does normalization of $|\psi\rangle$ carry in quantum mechanics?
5. How does the probability of an outcome relate to projection operators?

Tip:

For matrix exponentials and logarithms, understanding eigenvalues can simplify calculations significantly.