The question you've shared involves an operator $O_1$ in a three-dimensional Hilbert space $\mathcal{H}_3$, where you're asked to work with two different bases: the given orthonormal basis $\{ |e_1\rangle, |e_2\rangle, |e_3\rangle \}$ and the computational basis $\{ |c_1\rangle, |c_2\rangle, |c_3\rangle \}$. Let's address the parts of the question in order:

(a) Decompose $O_1$ over the computational basis

You are asked to decompose the operator $O_1$, given as: $O_1 = \frac{i}{\sqrt{2}} \left( |e_2\rangle \langle e_1| - |e_1\rangle \langle e_2| + |e_3\rangle \langle e_2| - |e_2\rangle \langle e_3| \right),$ into the computational basis $\{|c_1\rangle, |c_2\rangle, |c_3\rangle\}$.

1. Start by finding the matrix representations of $|e_1\rangle, |e_2\rangle, |e_3\rangle$ in terms of the computational basis. We are given: $|e_1\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \quad |e_2\rangle = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad |e_3\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}.$ Use these to write $O_1$ in matrix form.

2. Express the outer products like $|e_2\rangle \langle e_1|$, $|e_1\rangle \langle e_2|$, etc., using the computational basis. For instance: $|e_2\rangle \langle e_1| = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.$ Perform this for all terms in $O_1$.

3. Combine the terms and simplify the matrix for $O_1$ in the computational basis.

(b) Calculate the eigenvalues $\{\lambda_i\}$ and normalized eigenvectors $\{| \lambda_i \rangle\}$

1. Find the matrix of $O_1$ (from part (a)) and calculate its eigenvalues and eigenvectors.

2. Verify the closure relation $\sum_{j=1}^{3} \lambda_j | \lambda_j \rangle \langle \lambda_j | = \mathbb{I}_3$, where $\mathbb{I}_3$ is the identity matrix. This means checking that the eigenvalues and eigenvectors form a complete basis for the Hilbert space.

(c) Decompose $O_1$ over the basis $\{|\lambda_1\rangle, |\lambda_2\rangle, |\lambda_3\rangle\}$

Using the normalized eigenvectors found in part (b), express $O_1$ in this eigenbasis. This is essentially rewriting $O_1$ as a sum of projection operators along the eigenvectors, scaled by the corresponding eigenvalues.

Would you like me to work out the decomposition and eigenvalue calculations explicitly, or guide you through any specific part in more detail?

Here are 5 related questions for further exploration:

1. How is the matrix of an operator affected by changing the basis?
2. What is the significance of eigenvectors and eigenvalues in quantum mechanics?
3. How does the closure relation help in proving that the eigenbasis is complete?
4. How do outer products like $|e_i\rangle \langle e_j|$ represent transitions between states?
5. How would the solution change if $O_1$ were a Hermitian operator?

Tip: When decomposing an operator, always ensure that you express both the operator and the basis vectors in the same reference frame (basis) for consistency.