QM States on a Ring

Students should be able to:

1. (QmRingCompare)

   Before you begin, recall that an arbitrary state $\left|\Phi\right\rangle$ can be written in the $L_z$ eigenbasis as

   $$ \left| \Phi\right\rangle \doteq \begin{pmatrix} \vdots \\ \langle 2|\Phi\rangle \\ \langle 1|\Phi\rangle \\ \langle 0|\Phi\rangle \\ \langle -1|\Phi\rangle \\ \langle -2|\Phi\rangle \\ \vdots \end{pmatrix} = \begin{pmatrix} \vdots \\ a_{2} \\ a_{1} \\ a_{0} \\ a_{-1} \\ a_{-2} \\ \vdots \end{pmatrix} $$

   For this question, you will carry out calculations on each of the following normalized quantum states on a ring:

   $$ \left| \Phi_a\right\rangle = \sqrt{ 4\over 15}\left| 4\right\rangle + \sqrt{ 1\over 15}\left| 2\right\rangle +\sqrt{ 4\over 15}\left| 1\right\rangle +\sqrt{ 3\over 15}\left| 0\right\rangle +\sqrt{ 1\over 15}\left| -3\right\rangle +\sqrt{ 2\over 15}\left| -4\right\rangle $$

   $$ \left| \Phi_b\right\rangle \doteq \begin{pmatrix}\vdots \\ \sqrt{ 4\over 15} \\ 0 \\ \sqrt{ 1\over 15} \\ \sqrt{ 4\over 15} \\ \sqrt{ 3\over 15} \\ 0 \\ 0 \\ \sqrt{ 1\over 15} \\ \sqrt{ 2\over 15} \\ \vdots \end{pmatrix} $$

   $$ \Phi_c(\phi) = \sqrt {1\over {30 \pi}} \left( \sqrt{4} \left(e^{i 4 \phi} +e^{i \phi}\right) +\sqrt{3} + \sqrt{2} e^{-i 4 \phi} + e^{i 2 \phi}+e^{-i 3 \phi} \right) $$

   For each question state the postulate(s) of quantum mechanics you use to complete the calculation and show explicitly how you use the postulates to answer the question.

   1. If you measured the $z$-component of angular momentum for each state, what is the probability that you would obtain $4\hbar$? 0? $-2\hbar$?

   2. If you measured the energy for each state, what is the probability that you would obtain $0$? $\frac{\hbar^2}{2 I}$? $\frac{16 \hbar^2}{2 I}$? $\frac{25 \hbar^2}{2 I}$?

   3. How are the calculations you made for the different state representations similar and different? In a short paragraph, compare and contrast the calculation methods you used for each of the different representations (ket, matrix, wavefunction).

   4. If you measured the $z$-component of angular momentum, what other possible values could you obtain with non-zero probability?

   5. If you measured the energy, what other possible values could you obtain with non-zero probability?

2. (QmRingFunc)

   Consider the following normalized quantum state prepared for a particle on a ring of constant radius $r_0 =1$ at $t = 0$:

   $$\Phi(\phi)=\sqrt{8\over3 \pi } \sin^{2}\left( 3\,\phi \right)\cos \left( \phi \right)$$

   1. If you measured the $z$-component of angular momentum, what is the probability that you would obtain $\hbar$? $-3\hbar$? $-7\hbar$?

   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, what is the probability that you would obtain ${\hbar^2 \over 2 I}$? ${4\hbar^2 \over 2 I}$? ${25\hbar^2 \over 2 I}$?

   4. If you measured the energy, what possible values could you obtain with non-zero probability?

   5. What is the probability that the particle can be found in the region $0<\phi< {\pi \over 4}$? In the region ${\pi \over 4}<\phi< {3 \pi \over 4}$?

   6. Plot this wave function.

   7. What is the expectation value of $L_z$ in this state?

3. (QmRingNorm)

   Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy $$ \sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1 $$