(Solved): How to solve step-by-step? 4. Consider a system with l=1 whose hamiltonian is H=0(Lx2 ...

4. Consider a system with $l=1$ whose hamiltonian is $H=\frac{{\omega }_0}{\hslash }\left(L_x^2-L_y^2\right)$ where ${\omega }_0$ is a constant. Let be the basis $\{|-1⟩,|0⟩,|+1⟩\}$ formed by the eigenstates of $L_z$ with eigenvalues $m=-\hslash ,0,\hslash$. a) Write the matrix representing $H$ in the given basis. Find the steady states of the system and the corresponding energy values. b) At time $\mathrm{t}=0$ the state of the system is $|\psi (0)⟩=\frac{1}{\sqrt{2}}(|+1⟩-|-1⟩).$ Determine the state vector $|\psi (t)⟩$ at time t. If $L_z$ is measured at that time, what values can be obtained and with what probability?