7A state of a quantum system is described by a vector in a two-dimensional Hilbert space. The system Hamiltonian has distinct eigenvalues E₁ and E₂ which correspond to eigenvectors E₁) and E₂). A Hermitian operator 2 has distinct eigenvalues w₁ and ₂ corresponding to eigenvectors \left|\omega_{1}\right\rangle=\left(\left|E_{1}\right\rangle+\left|E_{2}\right\rangle\right) / \sqrt{2} \text { and }\left|\omega_{2}\right\rangle=\left(\left|E_{1}\right\rangle-\left|E_{2}\right\rangle\right) / \sqrt{2} A measurement of ohm was performed at time t = 0and yielded w₁.

1) Write the propagator for this system.

2) What is the expectation value of at a later time t?