28. Two Level Atom with Absorption and Emission:
(a) Use the qubit representation
$|e\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, excited state , $|g\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ ground state
(3.5.3)
CHAPTER 3. MATH BACKGROUND
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to create a matrix representation of $\sigma_z$, $\sigma_+$, $\sigma_-$, and $H = \frac{\hbar \omega_{eg}}{2}\sigma_z + \hbar \gamma (\sigma_+ + \sigma_-)$ via
the outer-product method.
b) Solve the eigenvalue problem for H with eigen-energies and eigenvectors.
29. Operator Algebra: Given the operators $\hat{A} = |0\rangle\langle 1| + |1\rangle\langle 0|$ and $\hat{B} = |0\rangle\langle 0| - |1\rangle\langle 1|$:
(a) Calculate $\hat{A}^2$ and $\hat{B}^2$.
(b) Prove that $\hat{A}\hat{B} + \hat{B}\hat{A} = 0$.
30. Projection Operator for a Qubit State:
Consider a qubit in the state $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$, where $\alpha$ and $\beta$ are complex numbers
satisfying $|\alpha|^2 + |\beta|^2 = 1$. Construct the projection operator $P_{\psi}$ that projects any state
onto $|\psi\rangle$, i.e.
$P_{\psi} = |\psi\rangle\langle \psi|$.
Show that
$P_{\psi}^2 = P_{\psi}$
(3.5.4)
(3.5.5)
