WKB approximation

From Academic Kids

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 \left( \frac{1}{2m[E-V(x)]} \right)^{1/4} \exp\left(\pm \frac{i}{\hbar} \int_{-\infty}^x dx' \sqrt{2m(E-V(x))}\right)<math>

The approximation is good when the potential, V(x), is slowly varying in comparison to the wavelength of the particle.

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.

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, Jeffreys 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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