Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
1. << Time Evolution Refresher (Mini-Lecture) | Quantum Ring Sequence | Visualization of Quantum Probabilities for a Particle Confined to a Ring >>
In this activity, your group will carry out calculation on the following quantum state on a ring: \begin{equation*} \left|{\Phi}\right\rangle =\sqrt{\frac{2}{3}}\left|{-3}\right\rangle +\frac{1}{\sqrt{6}}\left|{-1}\right\rangle +\frac{i}{\sqrt{6}}\left|{3}\right\rangle \end{equation*}
- Imagine you carry out a measurement to determine the \(z\)-component of the angular momentum of the particle at time, \(t\). Calculate the probability that you measure the \(z\)-component of the angular momentum to be \(3\hbar\). What representation/basis did you use to do this calculation and why did you use this representation?
- Imagine you carry out a measurement to determine the energy of the particle at time, \(t\). Calculate the probability that you measure the energy to be \(\frac{9\hbar^2}{2I}\). What representation/basis did you use to do this calculation and why did you use this representation?
- Calculate the probability that the particle can be found in the region \(0<\phi<\frac{\pi}{3}\) at some time, \(t\). What representation/basis did you use to do this calculation and why did you use this representation?
- Under what circumstances do measurement probabilities change with time?
Probability v. Probability Density: Students struggle with the two different ways of finding probability: for discrete and continuous measurements. Most recognize that they need to do an integral for a continuous quantity, but are not sure when to square (before integration or after).
\(\left|\int \phi_n^*(x)\Psi(x,t) dx\right|^2\) vs. \(\int\left|\Psi(t) \right|^2 dx\)
In particular, many students will forget to do the squaring for the calculation on the left because \(\int \phi_n^*(x) \Psi(x,t) dx\) looks a lot like \(\int \Psi^*(x,t) \Psi(x,t) dx\).
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 another activity and/or homework from the Quantum Ring Sequence that allows students to practice this more general method.
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy
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.group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition
Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.format_list_numbered Sequence
group Small Group Activity
30 min.
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.
magnetic fields current Biot-Savart law vector field symmetry
Students work in groups of three to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
group Small Group Activity
30 min.
electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula
Students work in groups of three to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(V(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
group Small Group Activity
30 min.
compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry
Students work in groups of three to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
computer Mathematica Activity
30 min.
central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.