Brief science items, end of October edition

Here are a few sciencey things I've been meaning to mention, while real life has been very busy.

• 

  Density of states for the first two 2D subbands

  Here is a cute html5 demo I made about densities of states, quickly coded with the help of gemini.  In statistical mechanics and solid state physics, it can be very helpful to try to count up the number of states (perhaps these are allowed electronic states, like electrons confined to a box or a lattice; perhaps these are electromagnetic modes in a cavity, as in the blackbody radiation problem; perhaps these are standing acoustic waves, as in the Debye vibrational heat capacity problem).  Imposing boundary conditions on the waves in our system picks out discrete allowed values of wavelengths (or equivalently \(\mathbf{k}\)-vectors).  Given those, "zoom out" so that we treat \(\mathbf{k}\) as a continuous variable, and toss in a dispersion relation that relates energy \(\epsilon\) to \(\mathbf{k}\), and you can find a density of states \(g(\epsilon) d\epsilon\), which tells you how many energy levels there are per unit energy between energies \(\epsilon\) and \(\epsilon + d\epsilon\).  For nonrelativistic electrons, with \(\epsilon \sim k^{2}\), in purely 1 spatial dimension \(g(\epsilon) \sim 1/\sqrt{\epsilon}\):  energy levels get farther and farther apart as you go up in energy, a result familiar from the particle-in-a-box.  Add in more 1D channels, and the total \(g\) looks like a sum of these 1D contributions.  Work in 2D, and you find that \(g(\epsilon) ~\)constant, independent of energy.  Work in 3D, and you find that \(g(\epsilon) \sim \sqrt{\epsilon}\).  This little applet lets you play with this, actually calculating the energies of the first 50000 states in a box whose dimensions you control.  You can see how the energy levels are distributed, and how that compares to the analytic expectations from above.   
  
• In the same spirit as this, here is a neat site that I hadn't seen until now, all about cameras and lenses, complete with interactive demonstrations.  This is just one of a wonderful collection of similar conceptual demos.  Seriously, these are great.  
• I also wanted to point out this very nice memorial tribute to Tony Leggett, who was an extremely nice, gracious person as well as an outstanding scientist.  Its publication coincided with a conference at UIUC in his honor.
• Belatedly, but in the same spirit, I've been wanting for some time to introduce my readers to Arianna Rosenbluth, a remarkable undergraduate alumna from Rice's physics department (NYT obituary here).  She deserves to be better known.  After her undergrad days, when incidentally she was the youth fencing champion of Houston (no gender distinction - she beat everyone, and qualified for the '44 and '48 Olympics, but didn't attend), she was a grad student of van Vleck at Harvard, in the group at the same time as Phil Anderson and Thomas Kuhn.  She was greatly responsible for the development of what became known as the Metropolis algorithm.  Back when I was department chair I was fortunate enough to correspond a tiny bit with her through her son, which was great.