assignment Homework
Central Forces 2023 (3 years)
The following are 3 different representations for the \(\textbf{same}\) state on a quantum ring for \(r_0=1\)
\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}
\begin{equation}
\left \Phi_b\right\rangle \doteq \left( \begin{matrix}
\vdots \\
i\sqrt{\frac{ 2}{12}}\\
0 \\
\sqrt{\frac{ 1}{12}} \\
\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}} \\
i\sqrt{\frac{ 2}{12}}\\
0 \\
\sqrt{\frac{4}{12} }\\
\vdots
\end{matrix}\right)
\begin{matrix}
\leftarrow m=0
\end{matrix}
\end{equation}
\begin{equation}
\Phi_c(\phi) \doteq \sqrt{\frac{1}{24 \pi}} \left( i\sqrt{2}e^{i 3 \phi} e^{i\phi}
+\sqrt{3}e^{i\frac{\pi}{4}} i \sqrt{2} e^{i\phi} + \sqrt{4}e^{i 3 \phi} \right)
\end{equation}

With each representation of the state given above, explicitly calculate the probability that \(L_z=1\hbar\). Then, calculate all other nonzero probabilities for values of \(L_z\) with a method/representation of your choice.

Explain how you could be sure you calculated all of the nonzero probabilities.

If you measured the \(z\)component of angular momentum to be \(3\hbar\), what would the
state of the particle be immediately after the measurement is made?

With each representation of the state given above, explicitly calculate the probability that \(E=\frac{9}{2}\frac{\hbar^2}{I}\). Then, calculate all other nonzero probabilities for values of \(E\) with a method of your choice.
If you measured the energy of the state to be \(\frac{9}{2}\frac{\hbar^2}{I}\), what would the state of the particle be immediately after the measurement is made?