The following are 2 different represtations for the \(\textbf{same}\) state on a quantum ring \begin{equation} \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{equation} \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}
Write down the matrix representation for the same state.
With all 3 representations, calculate the probability that a measurement of \(L_z\) will yield \(0\hbar\), \(-2\hbar\), \(2\hbar\).
If you measured the \(z\)-component of angular momentum to be \(2\hbar\), what would the state of the particle be immediately after the measurement is made?
What is the probability that a measurement of energy, \(E\), will yield \(0\frac{\hbar^2}{I}\)?,\(2\frac{\hbar^2}{I}\)?,\(4\frac{\hbar^2}{I}\)?
If you measured the energy of the state to be \(2\frac{\hbar^2}{I}\), what would the state of the particle be immediately after the measurement is made?