Langer correction
The Langer correction, named after the mathematician Rudolf Ernest Langer who developed it in 1937, is a correction to the WKB approximation for problems with radial symmetry. == Description == === In 3D systems === When applying WKB approximation method to the radial Schrödinger equation, − ℏ 2 2 m d 2 R ( r ) d r 2 + [ E − V eff ( r ) ] R ( r ) = 0 , {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}R(r)}{dr^{2}}}+[E-V_{\textrm {eff}}(r)]R(r)=0,} where the effective potential is given by V eff ( r ) = V ( r ) − ℏ 2 ℓ ( ℓ + 1 ) 2 m r 2 {\displaystyle V_{\textrm {eff}}(r)=V(r)-{\frac {\hbar ^{2}\ell (\ell +1)}{2mr^{2}}}} ( ℓ {\displaystyle \ell } the azimuthal quantum number related to the angular momentum operator), the eigenenergies and the wave function behaviour obtained are different from the real solution.