The WKB (Wentzel-Kramers-Brillouin) approximation is a semi-classical method used in quantum mechanics to approximate the solutions to the Schrödinger equation for slowly varying potentials. It is based on the assumption that the potential varies slowly enough that the wave function can be considered as a slowly varying amplitude modulated by a rapidly varying phase.
A slowly varying potential is one that changes gradually over space or time. In the context of the WKB approximation, this means that the potential changes slowly enough that the wave function can be considered as a slowly varying amplitude modulated by a rapidly varying phase.
The WKB approximation involves using a series expansion to approximate the wave function in terms of a slowly varying amplitude and a rapidly varying phase. This allows for a simpler and more manageable solution to the Schrödinger equation for slowly varying potentials.
The WKB approximation is most accurate for potentials that change slowly over space or time. It becomes less accurate for rapidly varying potentials, and may not be applicable at all for highly oscillatory potentials or potentials with sharp discontinuities.
The WKB approximation is commonly used in situations where the potential varies slowly over space or time, such as in the study of atomic and molecular physics, quantum optics, and solid state physics. It is also used in the study of semiclassical mechanics, where it provides a bridge between classical and quantum mechanics.