Langer correction
The Langer correction is a correction when the WKB approximation method is applied to three-dimensional problems with spherical symmetry.
When applying WKB approximation method to the radial Schrödinger equation
where the effective potential is given by
the eigenenergies and the wave function behaviour obtained are different from the real solution.
In 1937, Rudolph E. Langer suggested a correction
which is known as Langer correction or Langer replacement. This is equivalent to inserting a 1/4 constant factor whenever ℓ(ℓ + 1) appears. Heuristically, it is said that this factor arises because the range of the radial Schrödinger equation is restricted from 0 to infinity, as opposed to the entire real line.
By such a changing of constant term in the effective potential, the results obtained by WKB approximation reproduces the exact spectrum for many potentials.
That the Langer replacement is correct follows from the WKB calculation of the Coulomb eigenvalues with this replacement which reproduces the well known result. An even more convincing calculation is the derivation of Regge trajectories (and hence eigenvalues) of the radial Schrödinger equation with Yukawa potential by both a perturbation method (with the old factor) and independently the derivation by the WKB method (with Langer replacement)-- in both cases even to higher orders. For the perturbation calculation see Müller-Kirsten and for the WKB calculation Boukema.
Note that for 2D ststems, as the potential takes the form:
Langer correction goes:
...
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