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In physics, the WKB approximation, also known as WKBJ approximation, is the most familiar example of a semiclassical calculation in quantum mechanics in which the wavefunction is approximated by
- <math>\psi(x)\approx \frac{1}{\sqrt{2m[E-V(x)]}}\exp\left(\pm \frac{i}{\hbar} \int_{-\infty}^x dx' \sqrt{2m(E-V(x))}\right)<math>
The phase is simply determined by the condition that the rate of its change is fully determined by the energy E of the particle. The WKB approximation is essentially equivalent to one-loop Feynman diagrams in quantum field theory.
Basically, the observation is in a Feynman diagram, each internal vertex contributes <math>1/\hbar<math> whereas each propagator contributes <math>\hbar<math>. So, in a connected Feynman diagram with k external vertices, the contribution involves some integer power of <math>\hbar<math> which is greater than or equal to k-1. So, (in the small <math>\hbar<math> limit), at least perturbatively, the leading contribution to the k-point connected correlation function is of the order <math>\hbar^{k-1}<math>. This is the "classical" contribution. The <math>\hbar^k<math> contribution is the one-loop contribution, because simple graph theory tells us the "classical" contribution comes from tree diagrams while the "semiclassical" contribution comes from one-loop diagrams. Actually, this isn't technically precise. The semiclassical approximation really involves the next to leading order contributions to the one particle irreducible correlation functions. So, a feynman diagram with multiple cycles which are vertex disjoint contributes to the semiclassical approximation.
The approximation is good when the potential, V(x), is slowly varying in comparison to the wavelength of the particle.
This method is named after physicists Wentzel, Kramers, and Brillouin, who all developed it in 1926. In 1923, mathematician Harold Jeffreys had developed a general method of approximating linear, second-order differential equations, which includes the Schrödinger equation. But since the Schrödinger equation was developed two years later, and Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, Jeffereys is often neglected credit. Early texts in quantum mechanics contain any number of combinations of their initials, including WBK, BWK and BWKJ.
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