Quantum Mechanics Toys

Costas Papaliolios devised a little gadget for illustrating some quantum mechanical properties. It consists of a set of 21 hollow cubes with polarizing filters in them, labeled with Dirac's bra-ket notation. You know that if you use a polarizing filter to get light polarized vertically, another vertical filter will let the light through but a horizontal filter will block it (almost--the filters aren't perfect). But suppose you use a circularly polarizing filter instead? Some light will come through. And then if you put a horizontal filter after that, some light gets through.

But wait--didn't you just block all the horizontally polarized light with the first filter? Yes indeed, but then you re-measured with something that was a combination of both horizontal and vertical, and what gets through that is going to be, again, a mix of both horizontal and vertical. Measurements do not always commute. Even when you think you're not disrupting the system.

While Schwinger represented the atomic measurements of the Stern-Gerlach experiment as a matrix, Papaliolios represented the two states using the polarization of light. The quantum toys were equivalent to the magnetic field of the Stern-Gerlach experiment and to Schwinger’s matrix. By reordering, adding, or omitting blocks, you could see the unique characteristics of calculations made with matrices.

Very clever setup. The article is fun too:

Students recognized Schwinger’s brilliance, yet found his lectures impenetrable. Theoretical physicist and science writer Jeremy Bernstein recalled taking Schwinger’s course in 1950: “Schwinger was, it turned out, trying out an entirely new formulation of the theory on us—the old one would have been hard enough—and since he lectured from memory questions were discouraged.... After a few weeks I was lost.”

If you want to make your own, this article has instructions at the end.