QM Ring Compare

  • dirac notation matrix notation wavefunction notation particle on a ring angular momentum probability energy notation fluency
  • Central Forces 2022

    Before you begin, recall that an arbitrary state \(\left|\Phi\right\rangle\) can be written in the \(L_z\) eigenbasis as \begin{equation} \left| \Phi\right\rangle \doteq \left( \begin{matrix} \vdots \\ \left\langle {2}\middle|{\Phi}\right\rangle \\ \left\langle {1}\middle|{\Phi}\right\rangle \\ \left\langle {0}\middle|{\Phi}\right\rangle \\ \left\langle {-1}\middle|{\Phi}\right\rangle \\ \left\langle {-2}\middle|{\Phi}\right\rangle \\ \vdots \end{matrix}\right) = \left( \begin{matrix} \vdots \\ a_{2} \\ a_{1} \\ a_{0} \\ a_{-1} \\ a_{-2} \\ \vdots \end{matrix} \right) \end{equation}

    For this question, you will carry out calculations on each of the following normalized quantum states on a ring: \begin{equation} \left|{\Phi_a}\right\rangle = \sqrt\frac{ 4}{15}\left|{4}\right\rangle + \sqrt\frac{ 1}{15}\left|{2}\right\rangle +\sqrt\frac{ 4}{15}\left|{1}\right\rangle +\sqrt\frac{ 3}{ 15}\left|{0}\right\rangle +\sqrt\frac{ 1}{15}\left|{-3}\right\rangle +\sqrt\frac{ 2}{15}\left|{-4}\right\rangle \end{equation} \begin{equation} \left| \Phi_b\right\rangle \doteq \left( \begin{matrix} \vdots \\ \sqrt\frac{4}{15} \\ 0 \\ \sqrt\frac{1}{15} \\ \sqrt\frac{4}{15} \\ \sqrt\frac{3}{15} \\ 0 \\ 0 \\ \sqrt\frac{1}{15} \\ \sqrt\frac{2}{15} \\ \vdots \end{matrix}\right) \begin{matrix} \color{\red}{\leftarrow m=0} \end{matrix} \end{equation} \begin{equation} \Phi_c(\phi) = \sqrt{\frac{2}{15\pi}} ~ e^{i4\phi} + \sqrt{\frac{1}{30\pi}} ~ e^{i2\phi} +\sqrt{\frac{2}{15\pi}} ~ e^{i\phi} + \sqrt{\frac{1}{10\pi}} + \sqrt{\frac{1}{30\pi}} ~ e^{-i3\phi} + \sqrt{\frac{1}{15\pi}} ~ e^{-i4\phi} \end{equation}

    For each of states, answer the following questions. Perform your calculations in the same representation as the given state and, if are giving any answers by inspection, throughly describe your reasoning.

    1. If you measured the \(z\)-component of angular momentum for each state, what is the probability that you would obtain \(4\hbar\)? \(0\hbar\)? \(-2\hbar\)?
    2. If you measured the \(z\)-component of angular momentum, what other possible values could you obtain with non-zero probability?
    3. If you measured the energy for each state, what is the probability that you would obtain \(\displaystyle \frac{0\hbar^2}{2 I}\)? \(\displaystyle \frac{\hbar^2}{2 I}\)? \(\displaystyle\frac{16 \hbar^2}{2 I}\)? \(\displaystyle \frac{25 \hbar^2}{2 I}\)?
    4. If you measured the energy, what other possible values could you obtain with non-zero probability?
    5. Which postulate(s) of quantum mechanics did you have to use to complete each of these calculations? Don't just list them by number, give a description of the postulate.
    6. How are the calculations you made for the different state representations similar and different? How is identical information presented in each of the representations? Why are the coefficients in \(\Phi_c\) different? In a short paragraph, compare and contrast the calculation methods you used for each of the different representations (ket, matrix, eigenfunction).