WKB approximation
Other often-used letter combinations include JWKB and WKBJ, where the "J" stands for Jeffreys.
This method is named after physicists Gregor Wentzel, Hendrik Anthony Kramers, and Léon Brillouin, who all developed it in 1926.
The Schrödinger equation itself was not developed until two years later, and Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, so Jeffreys is often neglected credit.
Early texts in quantum mechanics contain any number of combinations of their initials, including WBK, BWK, WKBJ, JWKB and BWKJ.
[2] Earlier appearances of essentially equivalent methods are: Francesco Carlini in 1817, Joseph Liouville in 1837, George Green in 1837, Lord Rayleigh in 1912 and Richard Gans in 1915.
[3][4] The important contribution of Jeffreys, Wentzel, Kramers, and Brillouin 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.
Generally, WKB theory is a method for approximating the solution of a differential equation whose highest derivative is multiplied by a small parameter ε.
The asymptotic scaling of δ in terms of ε will be determined by the equation – see the example below.
To leading order in ϵ (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as
Therefore, the smallest error achieved by the WKB method is at best of the order of the last included term.
the number nmax will be large, and the minimum error of the asymptotic series will be exponentially small.
The wavefunction can be rewritten as the exponential of another function S (closely related to the action), which could be complex,
It is evident in the denominator that both of these approximate solutions become singular near the classical turning points, where E = V(x), and cannot be valid.
The integration in this solution is computed between the classical turning point and the arbitrary position x'.
For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions.
For most values of E, this matching procedure will not work: The function obtained by connecting the solution near
to the classically allowed region will not agree with the function obtained by connecting the solution near
Given that we expect wavefunctions to be of the following form, we can calculate their coefficients by connecting the different regions using Airy and Bairy functions.
in the given example shown by the figure, we require the exponential function to decay for negative values of x so that wavefunction for it to go to zero.
in the given example shown by the figure, we require the exponential function to decay for positive values of x so that wavefunction for it to go to zero.
In particular, the Maslov-corrected Bohr–Sommerfeld energies are good approximations to the actual eigenvalues of the Schrödinger operator.
The WKB wavefunction at the classical turning point away from it is approximated by oscillatory sine or cosine function in the classically allowed region, represented in the left and growing or decaying exponentials in the forbidden region, represented in the right.
The probability that the quantum particle will be found in the classically forbidden region is small.
This observation accounts for the peak in the wave function (and its probability density) near the turning points.
Since the rigid walls have highly discontinuous potential, the connection condition cannot be used at these points and the results obtained can also differ from that of the above treatment.
In 3 dimensions with spherically symmetry, the same condition holds where the position x is replaced by radial distance r, due to its similarity with this problem.
This result is also consistent with the use of equation from bound state of one rigid wall without needing to consider an alternative potential.
where the wavefunction in the classically forbidden region is the WKB approximation but neglecting the growing exponential.
This is a fair assumption for wide potential barriers through which the wavefunction is not expected to grow to high magnitudes.
