L20.2 The one-dimensional analogy for phase shifts
PROFESSOR: I'm going to write this e to the ikz somewhat differently so that you appreciate more what it is. So e to the ikz, I'll write it as square root of 4 pi over k. You say, where does that k come from? We'll see in a second. Sum over l, square root of 2l plus 1, i to the l, yl0, 1 over 2i. Basically what I'm going to do is I'm expanding this jl function for large x. So I'm going to take this equation and I'm going to expand it so that the argument is large, which means r is large. I want to describe for you those waves that are really going on here. So since the argument is kr and you have a kr here, that's the origin of the k that I pulled out in here. And then you have e to the i kr minus l pi over 2 over r, minus e to the minus i kr minus l pi over 2 over r. And this is valid for r much bigger than a. It's an approximate thing, because we approximated the Bessel function. This is exact. This equation is exact for all r. But th...