Cheap classroom demonstrations of quantum mechanics In a few weeks I'm going to be teaching a week-long class on quantum mechanics at a summer program for very mathematically talented high school students. I'm planning to focus on the finite-dimensional case, with maybe a short discussion of position and momentum at the end. The students will have taken linear algebra before showing up.
The last time I taught this I wished I had some physical demonstrations of some of the ideas from the class, so that's what I'm asking for here: inexpensive things I could show off in the classroom or in office hours to show students that the universe really does behave the way I'm insisting it does at the board. I'm willing to spend at most a couple hundred US dollars on this, maybe more for something especially cool.
Two ideas I already have are (a) playing around with light going through sequences of polarization filters, and (b) shining a laser pointer through two slits. I bet there are more! I'm especially interested in something that shows how measuring some quantum observable (like electron spin) can only yield one of a finite number of results.
 A: There are several different ways demonstrate quantum mechanics, depending on what your class is like and what you're trying to demonstrate.
Do what Einstein did
Black-body radiation is the light emitted by a hot object. Over a hundred years ago, Albert Einstein and Max Planck used black-body radiation and statistical mechanics to demonstrate that light is quantized into photons.
Under the Classical Theory of Radiation, every hot object emits the same color spectrum of light. When you increase the temperature of an object the quantity of light would increase but the color would stay the same.
But that's not what happens. When an object gets hotter its peak color changes. This is a result of statistical quantum mechanics. In fact, it was this experiment which helped us discover quantum mechanics in the first place, long before experiments with expensive things like lasers.
Different temperature objects emit different spectra because higher-frequency light has more energy per photon than lower-frequency light. Basically, an object can't emit any high-frequency light at all if its temperature is too low. Therefore changing an object's temperature changes the frequency distribution of its emission spectrum. If light wasn't quantized then every object could emit low amplitudes of high-frequency light.
So light a fire (or just buy a butane lighter). Examine how the hotter parts of the fire are blue and the colder parts of the fire are red. The color difference doesn't make sense unless light is quantized.
This kind of experiment shows how much scientists have discovered using cheap, simple tools.
Spectroscope
For a little more money you can buy an \$11 spectroscope. Use it to look at things with clear emission lines like a pure neon light or a computer screen displaying with a solid primary color rectangle. (It might take research, experimentation and additional purchases to find light sources with clear emission lines.) The light will split into clear lines instead of the continuums you'd expect from the previous black body experiment. These emission lines correspond to electrons jumping from one quantized energy level to another. In particular, they correspond to pairs of different values of $n$ among the electrons' quantum numbers.
This experiment is good because it lets students see the quantum numbers directly. If your students are good enough at math they might even be able to figure out which $n$ jumps the lines they look at correspond to.
A: A physical demonstration of quantum mechanics might not be the best route to follow. Quantum mechanics was discovered so late entirely because, on the macroscopic scale, it reduces to Newtonian mechanics. I would suggest that you look for some computer applications or videos which allow you to demonstrate, using images, the effects of quantum mechanics. These images will then appear to be macroscopic and will allow you to demonstrate things like quantum tunneling without having to just describe it with words. As @DanielSank pointed out, polarization and diffraction are not quantum phenomena. If I address your "...[I would like to] show students that the universe really does behave the way I'm insisting it does at the board" quote, I would simply say find examples. There are phenomena that are unexplainable without quantum mechanics. Perhaps find interesting graphics online as well to really illustrate what is going on. 
A: I just started reading "QED: The Strange Theory of Light and Matter" by Feynman.  In his first lecture, he talked about monochromatic light reflecting off a piece of glass of varying thicknesses and how the reflected amount varies with the thickness. An interesting example but may be too expensive. I suggest you read if you have access, his discussion of this in relation to QED. He wrote the book for a general audience. How does the second surface of the slide effect the reflection of the first surface? I rented the Kindle edition of the book for a month for a few dollars.
A: In addition to more active demonstrations of quantum mechanics, it may be interesting to point out things that we all take for granted which cannot be explained classically.
Just as a handful of examples, the optical properties of solids (e.g. why glass is transparent but bricks are not), the patterns and structure of the periodic table, and the existence and nature of atomic bonds (including, for example, the fact that ice is less dense than water) cannot be explained classically.  Beyond that, the rigidity of solid matter can be given a loose explanation in terms of electrostatic forces, but would not actually be possible without the additional influence of quantum mechanics.
