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PROFESSOR: So again, much of the difficulty here is just making sure you're doing the perturbation theory right and working with the two basis correctly. But now we're in the good position. We can combine our results. So here is what I want to combine. I want, in principle, to combine all the things, all the terms. And we've calculated all the terms to some degree. So we saw the Darwin term only affects l equals 0. On the other hand, spin orbit really requires s dot l. And the state should be acted nontrivially by l. If all your states are singlets, l gives 0 on them. So actually, you have a little bit of an uncomfortable situation because l acting on l equals 0 states will give you 0. But you still have the 1 over r cubed. And the 1 over r cubed has l dependencies here. So it's 0 in the denominator, 0 in the numerator. The whole spin orbit coupling doesn't seem to make sense for l equals 0. So most people say, physically, the spin orbit coupling vanis...

all right much of our study of feedback systems has really revolved around trying to prevent oscillations in feedback systems or even preventing getting very close to oscillation when we consider the relative stability of many of our systems today I'd like to look at the the converse of that problem in particular let's consider what happens when we wanted to want to design an oscillator and in fact a good quality sinusoidal oscillator an oscillator that provides very high purity sinusoidal signals as an example of describing function analysis we had looked at one function generator Lu which combined two Schmitt trigger and an integrator and we found out that that made a very convenient laboratory generator in that we were able to get simultaneously square waves and triangle waves and then by shaping the triangle wave very frequently manufacturers use that technique to build a oscillator or a function generator that can also provide sinusoidal signals the problem i...