assignment Homework

Working with Representations on the Ring

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}

1. 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.
2. Explain how you could be sure you calculated all of the non-zero probabilities.
3. 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?
4. 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.

5. 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?

assignment Homework

Paramagnet (multiple solutions)

Energy and Entropy 2021 (2 years) We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}

1. List variables in their proper positions in the middle columns of the charts below.

2. Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]

3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

   \[\left(\frac{\partial M}{\partial B}\right)_S \]

4. Evaluate your chain rule. Sense-making: Why does this come out to zero?