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. With all 3 representations, calculate the probability that a measurement of \(L_z\) will yield \(0\hbar\), \(-2\hbar\), \(2\hbar\).

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.