(Solved): 2. [Total: 22 pts] Consider a spin \( 1 / 2 \) particle with magnetic moment \( \vec{\mu}=-\frac{e ...

2. [Total: 22 pts] Consider a spin \( 1 / 2 \) particle with magnetic moment \( \vec{\mu}=-\frac{e}{m} \vec{S} \) in a uniform magnetic field \( \vec{B}=B_{o} \hat{z} \). a) Write the matrix representation of the Hamiltonian operator \( H \) in the \( S_{z} \) basis. b) Is \( |\psi\rangle=|+\rangle \) an eigenstate of \( \mathrm{H} \) ? Make a conceptual argument and then verify your argument with an explicit calculation. c) Is \( H \) compatible with \( S_{z} \) ? Verify by directly computing the commutator. d) What does your answer to the previous part tell you about the time dependence of the probability of measuring \( S_{z}=+\hbar / 2 \) ? e) Is \( H \) compatible with \( S_{x} \) ? Verify by directly computing the commutator. f) What does your answer to the previous part tell you about the time dependence of the probability of measuring \( S_{x}=+\hbar / 2 \) ?