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

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### 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.

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### 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.

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

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### 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.

Calculate ln(A) and Exp(iAt) for the matrix A = [[0, 1], [1, 0]] and analyze the quantum state |ψ> in Hilbert space H3 with observable Ô.

Explore matrix logarithms and exponentials for the matrix A = [[0, 1], [1, 0]] using diagonalization and series expansion. Analyze quantum state normalization, measurement outcomes, and probabilities in Hilbert space H3 with observable Ô. Suitable for university-level quantum mechanics and linear algebra students.

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