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Quantum Mechanics: WKB Approximation

By Dragica Vasileska1, Gerhard Klimeck2

1. Arizona State University 2. Purdue University

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Abstract

In physics, the WKB (Wentzel–Kramers–Brillouin) approximation, also known as WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) approximation, is the most familiar example of a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be slowly changing.

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, WKBJ and BWKJ.

Earlier references to the method are: Carlini in 1817, Liouville in 1837, Green in 1837, Rayleigh in 1912 and Gans in 1915. Liouville and Green may be called the founders of the method, in 1837.
The important contribution of Wentzel, Kramers, Brillouin and Jeffreys to the method was the inclusion of the treatment of turning points, connecting the evanescent and oscillatory solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a potential energy hill.

A thorough description of the WKB appproximation and its application in modeling nanoscale devices is given in the materials provided below.

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Researchers should cite this work as follows:

  • www.eas.asu.edu/~vasilesk
  • Dragica Vasileska; Gerhard Klimeck (2008), "Quantum Mechanics: WKB Approximation," http://nanohub.org/resources/4992.

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In This Series

  1. Slides: WKB Approximation 2

    09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, David K. Ferry

    www.eas.asu.edu/~vasileskNSF

  2. Slides: WKB Approximation 1

    09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, David K. Ferry

    www.eas.asu.edu/~vasileskNSF

  3. Reading Material: WKB Approximation

    09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska

    www.eas.asu.edu/~vasileskNSF

  4. Slides: WKB Approximation Applications

    09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, Gerhard Klimeck

    www.eas.asu.edu/~vasileskNSF

  5. Reading Material: Esaki Diode

    09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska

    www.eas.asu.edu/~vasileskNSF

  6. Homework: WKB Approximation

    09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, Gerhard Klimeck

    www.eas.asu.edu/~vasileskNSF

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