L15.3 Phase space and intuition for quantum adiabatic invariants
PROFESSOR: Let's do it biogeometrically. There's a nice geometric interpretation to this thing. So it comes from thinking of this in phase space. So this is motion in the x, p plain. So let's think of the motion of this oscillator in the x, p plane. Here, we have that the energy is equal to p squared over 2m plus 1/2 m omega squared x squared. So this is an ellipse in the x-- a closed orbit. A constant omega solution is an ellipse in this plane. That's because it's some something squared plus something squared with different coefficient. So here it is. It's some sort of ellipse like that. Semi-major axis, semi-minor axis. I actually don't know which is the major and which is a minor. But two semi axes. Well when p is equal to 0, what is the value of x defines this. So a is the value of x when p0-- so it's 2 square root of 2E over m omega squared. And b is the value of p when x is equal to 0. So it's just square root of 2mE. And here is ...