I’m giving an exam at 9:00 this morning– neither snow, nor more snow, nor blowing snow, nor single-digit temperatures shall stay the progress of shaping young minds. Anyway, to keep things lively while I’m proctoring the test, here’s a poll question inspired by the exam:
What’s your favorite calculational shortcut?
Today’s test is on basic quantum mechanics– photoelectric effect, Compton effect, the Bohr model of hydrogen, and simple solutions of the Schrödinger equation– and as such, features a lot of problems that are made easier by knowing some shortcut or another. Sometimes, these are numerical facts– for example, that Planck’s constant times the speed of light is 1240 ev-nm– and sometimes they’re mathematical techniques– such as knowing that an odd function integrated over a symmetric interval about the origin is zero– but there are a whole host of little facts that can dramatically shorten a problem, and reduce your chances of making a mistake.
So, what’s your favorite trick for making a long problem shorter?
For sheer labor-saving potential, I’d probably have to go with the odd/even function trick– there are a number of really horrible-looking integrals that come up when you do the sort of quantum mechanics that deals with actual wavefunctions, and many of them can be eliminated entirely by looking at the symmetry of the functions involved. You can arrive at the same answer by actually doing out the integral, of course, but it’s really easy to make a mistake– none of the students used the symmetry trick on a recent homework problem about finding expectation values for a gaussian wavepacket, and every one of them ended up getting a wrong answer.
I also get a lot of use out of hc=1240 eV-nm. It’s not that it’s all that difficult to remember the numerical values of Planck’s constant and the speed of light, but it takes a lot of problems from things that require a scientific calculator to work out down to arithmetic that I can easily do in my head, at least as long as I don’t have to convert back to joules…