Quantum Mechanics 1: Angular Momentum and Spin

In this problem, we're dealing with a situation where the magnetic field is at some angle relative to the z-axis, but we've written the Hamiltonian in the spin up spin down basis. Part A: Obtain the matrix representation of $H$ in the spin up spin down basis. Part B: Find the eigenvalues and normalized eigenvectors of $H$. Part C: Verify that the normalized eigenvectors of $H$, denoted as $E_1$ and $E_2$, satisfy the Dirac completeness relation.

Discuss the idea of normalization and derive the constraint on the coefficients for a general spin-1/2 ket state, ensuring that it is properly normalized. The constraint is given by: if $\left| \psi \right\rangle = a \left| + \right\rangle + b \left| - \right\rangle$, then $1 = |a|^2 + |b|^2$.