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 non-zero probabilities for values of \(L_z\) with a method/representation of your choice.
-
Explain how you could be sure you calculated all of the non-zero 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 non-zero 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?