Consider the following normalized quantum state on a unit ring:
\begin{equation}
\Phi(\phi)=\sqrt\frac{8}{3 \pi r_0} \sin^{2}\left( 3\,\phi \right)\cos \left( \phi \right)
\end{equation}
Translate this state into eigenfunction, bra/ket, and matrix representations. Remember that you can use any of these representations in the following calculations.
What is the expectation value of \(L_z\) in this state?
The wave function and it's probability density are plotted below. (I have set \(r_0=1\) to make the plotting easier). What features of these graphs (if any) tell you the expectation value of \(L_z\) in this state?
What is the probability that the particle can be found in the region \(0<\phi<
\frac{\pi}{4}\)? Repeat your calculation in the region \(\frac{\pi}{4}<\phi< \frac{3 \pi}{4}\)?
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?
Central Forces 2023 (3 years)
Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a
quantum mechanical particle of mass \(\mu\) constrained to move on a
circle of radius \(r_0\), given by:
\begin{equation}
\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}
\end{equation}
where \(N\) is the normalization constant.
Find \(N\).
Plot this wave function.
Plot the probability density.
Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
What is the expectation value of \(L_z\) in this state?
Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
Find the Fourier transforms
of \(f(x)=\cos kx\) and \(g(x)=\sin kx\).
Find the Fourier transform of \(g(x)\) using the formula for the Fourier transform of a derivative and your result for the Fourier transform of \(f(x)\). Compare with your previous answer.
In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function \(\tilde{f}(k)\) is a continuous histogram of how much each functions \(e^{ikx}\) contributes to the quantum state. What does the Fourier transform of the function \(\cos kx\) tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.
