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}}{2m}}{\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 or Langer replacement.^[1] 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.^[2] 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 $l(l+1)$ 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^[3] and for the WKB calculation Boukema.^[4]

References

↑ R.E. Langer, Phys. Rev. 51 (1937) 669.
↑ Harald J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. World Scientific (Singapore, 2012), p. 404.
↑ Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012), Chapter 16.
↑ J.I. Boukema, Physica 30 (1965) 1320, 1909.

The original publication.

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