WKB approximation

The semiclassical WKB approximation of quantum mechanics, which is named after Gregor Wentzel, Hendrik Kramers, and Léon Brillouin Anthony, provides an approximation of the solution of the one-dimensional stationary Schrödinger equation. The approximation is based on the assumption that the potential V (x ) 'slow' changes with the position and, therefore, is to find a solution from the constant potential.

The solution of the Schrödinger equation reads in this approximation

The two signs are for two independent solutions. They are only a good approximation when the potential on the extension of a wavelength changes only slowly.

History

The approximation in 1926 almost simultaneously and independently published by physicists Gregor Wentzel, Hendrik Kramers and Anthony Leon Brillouin in the framework of quantum mechanics, whose initials gave the island its name. They can also be found already in the works of various mathematicians and physicists such as Francesco Carlini (1817, in celestial mechanics ), George Green (1837 ), Joseph Liouville (1837 ), Lord Rayleigh (1912 ), Richard Gans (1915), Harold Jeffreys (1923 ) was used. It is therefore sometimes (in addition Jeffreys ) called WKBJ or Liouville - Green method. Werner Heisenberg also used the method in 1924 in his dissertation on hydrodynamics.

Derivation

From the one-dimensional stationary Schrödinger equation

Obtained at constant potential as a solution the plane wave

With. With a slow change of the potential, ie a potential which can be viewed in the order of the deBroglie wavelength to be constant, we can assume it and select a problem for constant-potential analog approach as follows.

Inserted into the Schrödinger equation is obtained

If no approximation has been made. We can develop in powers of now follows

This can put you in the Schrödinger equation:

Now you can calculate these terms to the desired order and collect after the power of.

Each to a power of associated term must then individually disappear.

For the second order is the Schrödinger equation:

For the differential equation in the zero-order term in

One can find a solution by

And it follows that

Implications for the transmission through a barrier

The WKB approximation is used to approximate non-rectangular barriers. To this end, the barrier is broken down into many thin rectangular portion barriers.

For the tunneling probability through this potential barrier, the individual tunnel probabilities are multiplied for each segment. This results in

In which

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