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.
1. << Time Dependence for a Quantum Particle on a Ring | Quantum Ring Sequence | Expectation Values for a Particle on a Ring >>
face Lecture
30 min.
assignment Homework
On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: https://phet.colorado.edu/sims/html/trig-tour/latest/trig-tour_en.html)
assignment Homework
In each of the following sums, shift the index \(n\rightarrow n+2\). Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:
assignment Homework
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.
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.
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.
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements
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.Use this Mathematica notebook or this GeoGebra applet to explore the time dependence of quantum states on a ring.
When students first open up Mathematica it is useful to walk them through the worksheet explaining the basic purpose of each step in the worksheet. It is not important to focus on the details at this point, but going through the steps in the worksheet helps keep some groups from getting stuck or just blasting through the worksheet without understanding what Mathematica is doing.
It may be helpful to review with students the meaning of the probability density since this is the primary quantity plotted in this worksheet.
The most common student mistake is to apply the time dependence to the wavefunction as a whole rather than applying the time dependence to each energy eigenfunction. Because these animations are premade, this activity will not help to address this issue unless you point to where and how the time dependence is added in the code.
After students have some time to play with the worksheet and understand what is being calculated and what value is being plotted, it is useful to ask some of the following questions.
Discussing what patterns students observed is a good start to a wrapup. This conversation can often lead to discussions of which wavefunctions result in observable time dependence and which do not. It is useful to look at the case of two wave functions added together, since this shows the simplest time dependence.
***Update this discussion*** This observation can be connected back to the time evolution of the two state system being sinusoidal with a frequency related to the energy difference between the two states.
\[|\Psi\rangle = a_m e^{-i {E_m\over\hbar} t} |m\rangle +a_n e^{-i {E_n\over\hbar}t} |n\rangle\]
\[\langle\phi|\Psi\rangle = a_m e^{-i {E_m\over\hbar}t} \langle\phi|m\rangle +a_n e^{-i {E_n\over\hbar}t} \langle\phi|n\rangle\]
\[\langle\phi|\Psi\rangle = e^{-i {E_m\over\hbar}t} (a_m \Phi_m(\phi) +a_n e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n(\phi))\]
\[|\langle\phi|\Psi\rangle|^2 = e^{i {E_m\over\hbar}t} (a_m^* \Phi_m^*(\phi) +a_n^* e^{i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)) e^{-i {E_m\over\hbar}t} (a_m \Phi_m(\phi) +a_n e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n(\phi))\]
\[|\langle\phi|\Psi\rangle|^2 = e^{i {E_m\over\hbar}t} e^{-i {E_m\over\hbar}t} (a_m^*a_m \Phi_m^*(\phi)\Phi_m(\phi)) + a_m^*a_n e^{i {(E_n-E_m)\over\hbar}t} \Phi_m^*(\phi)\Phi_n(\phi)) + a_n^*a_m e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)\Phi_m(\phi) + a_n^*a_n e^{i{(E_n-E_m)\over\hbar}t}e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)\Phi_n(\phi))\]
If \(a_m\) and \(a_n\) are real then
\[|\langle\phi|\Psi\rangle|^2 = a_m^2 |\Phi_m(\phi)|^2 + 2a_m a_n \cos\bigl(\frac{\Delta E_{mn}}{\hbar} t\bigr) \Phi_m(\phi)\Phi_n(\phi) + a_n^2 |\Phi_n(\phi)|^2\]
It it is sometimes helpful to have a follow up discussion after students have had some time to play with the worksheet outside of class. This gives each student some time to be in control of the worksheet and gives them time to play without the distractions and pressures of class.
This activity can be used as part of a sequence of Mathematica activities that allows one to explore probability densities for a particle first confined to a ring, then to the surface of a sphere, and finally for the entire three-dimensional hydrogen atom.
Related homework question: Particle on a Ring Time Dependence: Mathematica Notebook