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.
1. Quantum Ring Sequence | Superposition States for a Particle on a Ring >>
In this activity, your group will carry out calculations on each of the following normalized abstract quantum states on a ring: \begin{equation} \left|{\Phi_a}\right\rangle = \sqrt\frac{ 2}{12}\left|{3}\right\rangle + \sqrt\frac{ 1}{12}\left|{2}\right\rangle +\sqrt\frac{ 3}{12}\left|{0}\right\rangle +\sqrt\frac{ 2}{ 12}\left|{-1}\right\rangle +\sqrt\frac{ 1}{12}\left|{-3}\right\rangle +\sqrt\frac{ 3}{12}\left|{-4}\right\rangle \end{equation} \begin{equation} \left| \Phi_b\right\rangle \doteq \left( \begin{matrix} 0 \\ \sqrt\frac{ 2}{12}\\ \sqrt\frac{ 1}{12} \\ 0 \\ \sqrt\frac{ 3}{12} \\ \sqrt\frac{ 2}{12}\\ 0 \\ \sqrt\frac{1}{12} \\ \sqrt\frac{3}{12} \\ \end{matrix}\right) \end{equation} \begin{equation} \Phi_c(\phi) \doteq \sqrt {\frac{1}{24 \pi r_0}} \left( \sqrt{2}e^{i 3 \phi} +e^{i 2\phi} +\sqrt{3} + \sqrt{2} e^{-i 1 \phi} + e^{-i 3 \phi}+\sqrt{3}e^{-i 4 \phi} \right) \end{equation}
For each question state the postulate of quantum mechanics you use to complete the calculation and show explicitly how you use the postulates to answer the question.
- For each state above, what is the probability that you would measure the \(z\)-component of angular momentum to be \(-4\hbar\)? \(0\hbar\)? \(-2\hbar\)? \(3\hbar\)?
- What other possible values for the \(z\)-component of angular momentum could you have obtained with non-zero probability?
- For each state, what is the probability that you would measure the energy to be \(\displaystyle \frac{16\hbar^2}{2 I}\)? \(0\)? \(\displaystyle\frac{4 \hbar^2}{2 I}\)? \(\displaystyle \frac{9 \hbar^2}{2 I}\)?
- If you measured the energy, what other possible values could you obtain with non-zero probability?
- How are the calculations you made for the different state representations similar and different from each other? Be prepared to compare and contrast the calculations you made for each of the different representations (ket, matrix, eigenfunction).
This activity flows naturally from a lecture in which the eigenstates for energy and angular momentum on a ring are found. Many of the calculations done here are similar to calculations they have done before, but this activity emphasizes the different representations we use for quantum calculations and highlights when each representation is most useful.
Remind the students that that an arbitrary state \(|\Phi\rangle\) can be written in the \(L_z\) eigenbasis as
\[ \eqalign{\left| \Phi\right\rangle &\doteq \pmatrix{\vdots \cr \langle 2|\Phi\rangle \cr \langle 1|\Phi\rangle \cr \langle 0|\Phi\rangle \cr \langle -1|\Phi\rangle \cr \langle -2|\Phi\rangle \cr \vdots} = \pmatrix{\vdots \cr a_{2} \cr a_{1} \cr a_{0} \cr a_{-1} \cr a_{-2} \cr \vdots}} \]
Including this in the introduction to this activity should help students avoid confusion about the ordering of the elements in the column vectors used in this activity.
Students readily grasp the strategy of finding probability amplitudes “by inspection” when they are given an initial state written as a sum of eigenstates. We find that students then find it extremely difficult to find probability amplitudes of wavefunctions that are not written this way (i.e. using an integral to find the expansion coefficients of a function). This activity should be followed up with other activities in the Quantum Ring Sequence and homework that allow students to practice this more general method.
Associated Homework Problem: QM Ring Compare
assignment Homework
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}
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?
group Small Group Activity
30 min.
group Small Group Activity
30 min.
keyboard Computational Activity
120 min.
finite difference hamiltonian quantum mechanics particle in a box eigenfunctions
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then usenumpy
to solve for eigenvalues and eigenstates, which they visualize.
assignment Homework
group Small Group Activity
30 min.
face Lecture
5 min.
assignment Homework
Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.
You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}