The WKB method can be used to approximate the energy levels of simple quantum mechanical systems. In this lecture Prof. Strogatz shows three examples of how this is done for Schrödinger equations in 1-D. The payoff of the second and third examples, which involve two turning points, is a formula for the energy levels of a quantum particle moving in a potential well. The discrete set of allowed energies correspond to eigenvalues of the differential equation. The analysis predicts the phenomenon of "zero-point energy": even in its lowest energy state, a quantum system always fluctuates (even at a temperature of absolute zero!) and thus can never be completely at rest.
