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.

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 $|e_3\rangle$.

Follow-Up Questions and Tips

1. How do you calculate eigenvalues and eigenvectors for