Representations of Probabilities on the Ring

  • Central Forces 2021 \(\left|{\Phi}\right\rangle \) is a state for a quantum particle confined to a ring. \begin{equation} \left|{\Phi_a}\right\rangle = i\sqrt\frac{ 2}{12}\left|{3}\right\rangle - \sqrt\frac{ 1}{12}\left|{1}\right\rangle +\sqrt\frac{ 3}{12}e^{i\frac{\pi}{4}}\left|{0}\right\rangle -i\sqrt\frac{ 2}{ 12}\left|{-1}\right\rangle +\sqrt\frac{ 4}{12}\left|{-3}\right\rangle \end{equation}
    1. Plot a histogram for the probabilities of \(L_z\).

    2. Make an argument for what you think the expectation value of \(L_z\) will be and mark it with an X on the horizontal-axis of your histogram.

    3. Calculate the expectation value of \(L_z\).

    4. Do parts 1-3 again for energy.