WKB approximation
The WKB approximation is a method for solving the wave equation when the potential changes only slowly compared with the wavelength of the particle. The picture is that of a wave whose wavelength gently varies from place to place: where the particle has lots of kinetic energy it oscillates quickly, and where it slows down its waves stretch out. The method stitches together these locally simple wave shapes into a global solution.
It earns the name semiclassical because it sits halfway between quantum and classical physics. The phase of the wave accumulates as an integral of the local momentum along the path, which is exactly the classical action, while a slowly varying amplitude keeps probability properly conserved. In regions a classical particle could never enter, the wave does not oscillate but decays smoothly, which is precisely how the method estimates tunneling through a barrier.
WKB shines at two tasks. It gives a remarkably good rule for the allowed energies of a smooth potential well, by demanding that the wave fit neatly between the turning points. And it provides a clean estimate of the small probability of tunneling through a wide barrier, as an exponential of the integrated decay across the forbidden region. Its honesty is built in: it falters near the turning points, where the slow-variation assumption fails and special patching is needed.
Quantized energies follow from fitting the locally varying wave between the classical turning points.
WKB is an approximation, not exact: it breaks down where the potential varies rapidly and at the classical turning points, where the naive formula diverges and must be repaired by connection formulas.