Langer correction

The Langer correction is a correction when WKB approximation method is applied to three-dimensional problems with spherical symmetry.

When applying WKB approximation method to the radial Schrödinger equation

$-\frac{\hbar^2}{2 m} \frac{d^2 R(r)}{dr^2} + [E-V_\textrm{eff}(r)] R(r) = 0$

where the effective potential is given by

$V_\textrm{eff}(r)=V(r)-\frac{\ell(\ell+1)\hbar^2}{2mr^2}$

the eigenenergies and the wave function behaviour obtained are different from real solution.

In 1937, Rudolph E. Langer suggested a correction

$\ell(\ell+1) \rightarrow \left(\ell+\frac{1}{2}\right)^2$

which is known as Langer correction. 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.

References

• The original publication.