OK, if we really understand what is going on at this point, we ought to be able to get a rough cut at a system with three down spins.
So we've got a couple demonstrations that Feynman's solution is correct; what does this tell us about going forward?
[more ]In the first pass of verifying Feynman's solution, we took a moment at the very end to convert part of the expression from cosines to exponentials. Does solving the problem get any easier if we do that right away?
OK, so Feynman has provided a solution, and furthermore one that doesn't look particularly like the one used in the approximation. So the first step is to work the problem using his solution, so that we may understand how it works, and hopefully get some insight as to where it comes from.
[more ]In Feynman's lecture on the independent particle approximation (vol III, ch 15 - available in volume 4 of the audio lectures), he describes spin waves. Spin waves are one kind of boojum that can propagate through a crystal lattice.
The fun problem that he glosses over is the interaction of two spin waves in a single lattice. In the lecture he remarks that he was able to solve it, but didn't bring the solution with him. The text contains the exact solution in a parenthetical remark, with no further discussion of where the solution comes from.
So the next few posts will be describing my adventures in understanding what's going on.