Tuesday, February 12, 2013

tan(x)=x The Overlooked Approximation and Other Notes

My quantum professor always knows exactly the right approximations to make, (I suppose it helps to be the person who wrote the homework).  Regarding the recent homework questions involving boundary conditions in shallow square potential wells which are just oozing with tangents of ratios of wave numbers, he pointed out a trig approximation that's often overlooked, (at least by students like myself).  At small angles (picture 1)


the tangent of x is just x.

Yup, small angles aren't just for sine anymore!  The tangent of a small angle is also equal to it's argument.  I had to go through the additional little thought process of (picture 2)


Here's a graph of tan and sin overlayed with x so you can see what's going on. (picture 3)



The Levi-Civita Symbol
Another mathematical tool that has come up repeatedly in the last few weeks is the Levi-Civita symbol.  If you've studied the cross product or the vector curl you've seen it.  Wikipedia defines the symbol like this (picture 4)


They go on to say that it's +1 if the indices are an even permutation of (1,2,3), and -1 if the indices are an odd permutation of (1,2,3).  My rule for remembering this is that if you can read the indices in order from left to right, (wrapping when you hit a close paren), then the results is +1.  If you can read the indices in order from right to left, (wrapping when you hit an open paren), then the result is -1.  Positive numbers on the number line go from left to right and negative numbers go from right to left... Voila.

This symbol turns up over and over again and is very handy to memorize.  When you want to put things succinctly, and impress your friends, you can write down the components of a cross product, or a curl, or any number of electromagnetics equations using the Levi-Civita symbol.  For example, Wikipedia defines the vector curl  as (picture 5)



References:
1.  http://en.wikipedia.org/wiki/Levi_civita_symbol

2.  xkcd style graphs:
http://mathematica.stackexchange.com/questions/11350/xkcd-style-graphs


Picture of the Day:
Today's pictue of the day is taken from Wikipedia and is of Tullio Levi-Civita.  He's one of the few 19th and 20th century scientists I've researched lately who was actually smiling!  He's cool! (picture 6)



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