I am teaching our junior quantum mechanics out of Townsend’s book, which starts with Dirac notation and spin states, and when it lands in the position representation the students still haven’t made the full connection to the material they learned in their ‘modern physics’ course.

Here they are re-introduced to things like bound states (particle in a box so far mainly) and we read but did not discuss in detail a derivation of the tunneling coefficients. Yesterday was my designated ‘wrap-up the reintroduction/transition from Dirac notation to position space’ day. I chose to spend the class meeting time 1/2 on doing in-class simulations and 1/2 getting started or finishing the quantitative problems I had assigned. I found myself spending a big part of the afternoon discussing one particular problem with the students, which would perhaps have gone differently if I hadn’t (a) assigned it as a ‘mini-test’ (which meant that they couldn’t consult with anyone but me about it) or (b) gone over some material in class instead of the simulation. I am still a little puzzled about the number of visits I had, since things have gone differently with this section of the course with previous classes, but difference from previous years is a given, so I am just noting this and moving on. About the only thing I intend to change if I assign it in the same way in the future is to remember to ask everyone to look at the problems and make sure they knew how to start every one of them before they came to class.

I have assigned some version of the simulations as take-home assignment before, but we never got to discuss things together as a class, and I wasn’t available to guide their explorations. This time, by the end of the class I saw and heard some sense of understanding for some or many of my students about the qualitative and conceptual sense of time-dependent quantum mechanics at this level: How wave packets spread, in what sense they are superpositions of states and how you might move back and forth between the position and momentum representation, and how uncertainty connected with that, some sense of how different choices of superpositions of states affects the dynamics, how properties of time-evolving states are such that some expectation values are ‘stationary’ while others not, how even things like tunneling were built into the initial condition. We also discussed — very quickly, but more than once — other interesting notions like how symmetry affects things like energy eigenfunctions and eigenvalues, how tunneling splitting between states for a 2-well problem works, and how when generalized leads to band structure. It was possibly ambitious and maybe none of this landed or will stay or be useful but we shall see. I certainly walked away reminded of the importance of the transition — from introduction to the math to ‘understanding’ -in my grasp of most of these ideas. I do know that the difficulty of the transition was because the math was new and it took some time to translate into intuition. I learned all this in the world before computer simulations/movies/visualizations existed, and I am *sure* it would have gone better and faster for me by far if I had had access to such tools. Perhaps all of us teach the course we wish we had had.

Below is an adapted version of what I said in my pre-class note for my quantum mechanics class.

—

Tomorrow in class I’d like us to start with numerical experiments (quantum video games) on the position-representation quantum mechanics that we’ve just revisited. We will work on HW for the second 1/2 of the class.

We will play with some jnlp apps to improve our intuition.

If you spend 20 minutes or so tonight (or before class tomorrow) looking at the apps (links below), it would save an enormous amount of start-up time in class tomorrow. Just try to get a sense of what the various knobs and buttons on the apps do, and play around a bit with the settings. These apps should work with lab computers, and/or with your laptops. I just installed it on my office laptop.

Here are the basic ideas to explore

(a) Tunneling: This jnlp app allows you to ask all sorts of questions about tunneling: http://phet.colorado.edu/en/simulation/quantum-tunneling

(b) Bound states: http://phet.colorado.edu/en/simulation/bound-states; make sure you click over to the two-well and multi-well bound states; both to to get a sense of more complicated problems and also how band structure arises in quantum mechanics.

—

When they walked in to class, they found 10 questions that I thought they could explore and were free to choose what they wanted and how many they wanted as long as they kept working at the questions. We spent 20-25 minutes exploring these questions (I had checked that we had enough laptops in the class and they landed up being in pairs or threes), and then we talked for 10 minutes together as a class. We finished with them working on the problems assigned.