You're overthinking this. Yes, in principle we do known the fine structure constant to something like 9 orders of magnitude both experimentally and theoretically.

However, when would you use the pi = 3 approximation? Certainly not when you're in front of a computer, or if you were preparing some experimental results for publication. But, if you're in the lab and need to quickly make some calculations, or just to see if something is feasible and worth spending more time on, pi = 3 isn't so bad.

Example, measuring the fine structure. Sure, you can predict where these energy levels are supposed to be to probably whatever our error on knowing the mass of an electron is. And because you know the fine structure so precisely, you should be able to make a very accurate prediction on where that is. However, throw most of those digits out the door, because a lot will be hidden behind doppler broadening. So when you make your measurement in your fabry perot etalon, you'll probably make a precise measurement

http://en.wikipedia.org/wiki/File:Fabry_Perot_Etalon_Rings_F...

but how accurately can you really measure the position of those fringes? Sure, free spectral range probably lets you get down to about MHz region or so, but the doppler broadened linewidth is probably an order larger than that. Which brings us back to, you've got these experimental errors, why care about 9 digits of precision if you just want a quick and dirty calculation to get things set up?

Anyways, that's all he's trying to say. In most experiments, there will be some sort of experimental error hurting you. Be sloppy in the beginning just to get a feel for things.

Two aspects spring to mind. First, we don't know how long ago this course was, so it's quite possible some of the story is either dated or a slightly misremembered anecdote for a physicist who was brought up with slide rules for his calculations. Second, the quote specifies "experimentally", which implies measurements limited to two orders of magnitude, i.e., measured vs theoretical. Alas, while I have a few slipsticks in the closet, I am not now nor have I ever been a physicist, so I could be totally wrong on this.

> I can't think of a variable in quantum mechanics we "don't know to within 2 orders of magnitude".

Maybe the neutrino masses differences? In my intro QM class, we did a problem where we calculated the distance it took for one neutrino flavor to mix into another, which depends on the mass differences. You wouldn't want to take a square root of eV, though.

Or it could be it's one of the CKM angles. The latter are known to 1 part in 10^4, but only because of the relatively recent BaBar and BELLE experiments. Maybe back when this guy was an undergrad they only know them to 1 part in 100. Seems kinda advanced for 3rd semester of QM...