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The problem is about a quantum state in a Hilbert space \( \mathbb{H}_3 \) with a given state vector \( |\psi\rangle \) and an observable \( \hat{O} \) represented by a matrix. Let's go through each part:

1. **Problem Summary**:
   - The orthonormal basis of the Hilbert space \( \mathbb{H}_3 \) is \( \{ |e_1\rangle, |e_2\rangle, |e_3\rangle \} \).
   - The state vector \( |\psi\rangle \) is:
     \[
     |\psi\rangle = \frac{1}{2}|e_1\rangle + \frac{i \sqrt{3}}{2}|e_2\rangle - \frac{1}{\sqrt{8}}|e_3\rangle
     \]
   - The observable \( \hat{O} \) is represented by the matrix:
     \[
     \hat{O} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 2i & 0 \\ 0 & 0 & -2i \end{pmatrix}
     \]

2. **Questions**:
   - (a) **Show that \( |\psi\rangle \) is normalized**.
   - (b) **Find the possible measurement outcomes of \( \hat{O} \)**.
   - (c) **Calculate the probability of obtaining a specific measurement outcome (e.g., eigenvalue of 3)**.
   - (d) **Determine the post-measurement state if outcome 3 is recorded**.

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### Solutions:

#### (a) Show that \( |\psi\rangle \) is normalized.
To check normalization, we need to verify that \( \langle \psi | \psi \rangle = 1 \).

1. Write out \( |\psi\rangle \):
   \[
   |\psi\rangle = \frac{1}{2}|e_1\rangle + \frac{i \sqrt{3}}{2}|e_2\rangle - \frac{1}{\sqrt{8}}|e_3\rangle
   \]

2. Calculate \( \langle \psi | \psi \rangle \):
   \[
   \langle \psi | \psi \rangle = \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{\sqrt{8}}\right)^2
   \]
   This simplifies to:
   \[
   = \frac{1}{4} + \frac{3}{4} + \frac{1}{8} = 1
   \]
   Therefore, \( |\psi\rangle \) is indeed normalized.

#### (b) Find the possible measurement outcomes of \( \hat{O} \).
The possible measurement outcomes are the eigenvalues of \( \hat{O} \).

The matrix \( \hat{O} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 2i & 0 \\ 0 & 0 & -2i \end{pmatrix} \) is already in diagonal form, so its eigenvalues are the entries on the diagonal:
   - \( -1 \)
   - \( 2i \)
   - \( -2i \)

Thus, the possible measurement outcomes are \( -1 \), \( 2i \), and \( -2i \).

#### (c) Calculate the probability of obtaining each measurement outcome.
The probability of measuring an eigenvalue \( \lambda \) is \( |\langle \phi_\lambda | \psi \rangle|^2 \), where \( |\phi_\lambda\rangle \) is the eigenvector associated with \( \lambda \).

1. **Eigenvalue \( -1 \)**: Corresponds to the basis vector \( |e_1\rangle \).
   \[
   \text{Probability} = |\langle e_1 | \psi \rangle|^2 = \left| \frac{1}{2} \right|^2 = \frac{1}{4}
   \]

2. **Eigenvalue \( 2i \)**: Corresponds to the basis vector \( |e_2\rangle \).
   \[
   \text{Probability} = |\langle e_2 | \psi \rangle|^2 = \left| \frac{i \sqrt{3}}{2} \right|^2 = \frac{3}{4}
   \]

3. **Eigenvalue \( -2i \)**: Corresponds to the basis vector \( |e_3\rangle \).
   \[
   \text{Probability} = |\langle e_3 | \psi \rangle|^2 = \left| -\frac{1}{\sqrt{8}} \right|^2 = \frac{1}{8}
   \]

#### (d) Determine the state of the system after a measurement with outcome \( -2i \) is recorded.
If the measurement yields the outcome \( -2i \), the system will collapse to the eigenstate associated with \( -2i \), which is \( |e_3\rangle \).

Thus, after the measurement, the state of the system will be \( |e_3\rangle \).

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### Follow-Up Questions and Tips
1. How do you calculate eigenvalues and eigenvectors for

In the Hilbert space H_3, consider the orthonormal basis {e1, e2, e3}. Assume that the physical state of a given quantum system is represented by the state vector ψ = 1/2 |e1> + (i√3)/2 |e2> - 1/√8 |e3>. (a) Show that the state ψ is normalized. (b) Consider the observable Ô represented by the matrix Ô = [[-1, 0, 0], [0, 2i, 0], [0, 0, -2i]]. Find the possible measurement outcomes of the physical quantity embodied by Ô. (c) What is the probability that a measurement of Ô over the system prepared in state ψ gives the outcome 3? (d) What is the state of the system immediately after such a measurement is performed and outcome 3 recorded?

Explore a quantum mechanics problem involving a state vector in Hilbert space H_3. Learn how to show that a quantum state is normalized, find the possible measurement outcomes of an observable matrix, calculate the probability of a specific outcome, and determine the post-measurement state.

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