Langer correction

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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_{eff}(r)] R(r) = 0

where the effective potential is given by

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

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

In 1937, R.E. Langer suggested a correction

l(l+1) \rightarrow \left(l+\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.