Time Dependence on a Ring

Students should be able to:

The following types of problems can be done as part of in-class activities and/or as homework. For examples of in-class activities, see:

• Energy and Angular Momentum for a Particle Confined to a Ring
• Time Dependence for a Particle Confined to a Ring
• Superposition States for a Particle Confined to a Ring
• Expectation Values for a Particle Confined to a Ring

Even if done in-class, it can be helpful to have follow-up homework that helps to solidify additional points (such as degeneracy or probability in wave function notation).

1. (RingTable) Complete the table (EigenTableRingEmpty.pdf) which summarizes much of what you've learned about the eigenstates of the ring by filling in all the empty boxes.
2. (RingFunction) Some quantum calculations for a particle confined to a ring.

   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:

   $$\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}$$ where $N$ is the normalization constant.

   1. Find $N$.

   2. Plot this wave function.

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

3. (RingKet) Some quantum calculations in Bra-ket notation for a particle confined to a ring.

   Consider the normalized state $\left| \Phi\right\rangle$ for a quantum mechanical particle of mass $\mu$ constrained to move on a circle of radius $r_0$, given by:

   $$\left| \Phi\right\rangle= \frac{\sqrt 3}{2}\left| 3\right\rangle+ \frac{i}{2}\left| -2\right\rangle$$

   1. What is the probability that a measurement of $L_z$ will yield $2\hbar$? $3\hbar$?

   2. 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?

   3. What is the probability that a measurement of energy will yield $E=\frac{2\hbar^2}{I}$?

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

4. (RingCompare) Some quantum calculations in three different representations for a particle confined to a ring.

   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?

5. (RingTimeDep) Some quantum calculations for a particle confined to a ring at different times.

   In this problem, you will carry out calculations on the following normalized abstract quantum state on a ring:

   $$\left| \Psi\right\rangle = \sqrt{ 1\over 4} \left(\left| 1\right\rangle + \sqrt{2}\left| 2\right\rangle +\left| 3\right\rangle\right) $$

   1. You carry out a measurement to determine the energy of the particle at time $t=0$. Calculate the probability that you measure the energy to be $\frac{4 \hbar^2}{2 I}$.

   2. You carry out a measurement to determine the z-component of the angular momentum of the particle at time $t=0$. Calculate the probability that you measure the z-component of the angular momentum to be $3 \hbar$.

   3. You carry out a measurement on the location of the particle at time, $t=0$. Calculate the probability that the particle can be found in the region $0<\phi< \frac{\pi}{2}$.

   4. You carry out a measurement to determine the energy of the particle at time $t = \frac{2 I}{\hbar} \frac{\pi}{4}$. Calculate the probability that you measure the energy to be $\frac{4 \hbar^2}{2 I}$.

   5. You carry out a measurement to determine the z-component of the angular momentum of the particle at time $t = \frac{2 I}{\hbar}\frac{\pi}{4}$. Calculate the probability that you measure the z-component of the angular momentum to be $3 \hbar$.

   6. You carry out a measurement on the location of the particle at time, $t = \frac{2 I}{\hbar}\frac{\pi}{4}$.

      Calculate the probability that the particle can be found in the region $0<\phi< \frac{\pi}{2}$.

   7. Write a short paragraph explaining what representation/basis you used for each of the calculations above and why you chose to use that representation/basis.

   8. In the above calculations, you should have found some of the quantities to be time dependent and others to be time independent. Briefly explain why this is so. That is, for a time dependent state like $\left| \Psi\right\rangle$ explain what makes some observables time dependent and others time independent.

6. (RingFunc) Some quantum calculations on a wavefunction that is not trivially separated into energy eigenstates for a particle confined to a ring.

   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 ${\pi \over 4}<\phi< {3 \pi \over 4}$?