The following are 2 different representations for the same state on a quantum ring \begin{align} \left|{\Phi}\right\rangle = \sqrt{\frac{1}{2}}\left|{2}\right\rangle -\sqrt{\frac{1}{4}}\left|{0}\right\rangle +i\sqrt{\frac{1}{4}}\left|{-2}\right\rangle \end{align} \begin{equation} \Phi(\phi) \doteq \sqrt{\frac{1}{8 \pi r_0}} \left(\sqrt{2}e^{i 2 \phi} -1 + ie^{-i 2 \phi} \right) \end{equation}

1. Write down the matrix representation for the same state.

2. In each of the three representations, write the expressions you would use to evaluate the probabilities that a measurement of \(L_z\) will yield \(0\hbar\), \(-2\hbar\), and \(2\hbar\). Then choose one of the representations and carry out the probability calculations.

3. If you measured the \(z\)-component of angular momentum to be \(2\hbar\), write down the full resultant state immediately after the measurement.

4. If an energy measurement is performed on the state \(\Phi(\phi)\), what is the probability that the energy measurement will yield each of the following values: \(0\frac{\hbar^2}{I}\)?,\(2\frac{\hbar^2}{I}\)?,\(4\frac{\hbar^2}{I}\)?

5. If you measured the energy of the state to be \(2\frac{\hbar^2}{I}\), write down the full resultant state immediately after the measurement.