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The Langer correction, named after the mathematician Rudolf Ernest Langer, is a correction to the
WKB approximation In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
for problems with radial symmetry.


Description


In 3D systems

When applying WKB approximation method to the radial
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
, : -\frac \frac + -V_\textrm(r)R(r) = 0 , where the
effective potential The effective potential (also known as effective potential energy) combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the 'opposing' centrifugal potential energy with the potential energy of a ...
is given by :V_\textrm(r)=V(r)-\frac ( \ell the
azimuthal quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe t ...
related to the
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
), the eigenenergies and the wave function behaviour obtained are different from the real solution. In 1937, Rudolf E. Langer suggested a correction :\ell(\ell+1) \rightarrow \left(\ell+\frac\right)^2 which is known as Langer correction or Langer replacement. This manipulation is equivalent to inserting a 1/4 constant factor whenever \ell(\ell+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 the replacement which reproduces the well known result.


In 2D systems

Note that for 2D systems, as the effective potential takes the form :V_\textrm(r)=V(r)-\frac, so Langer correction goes: :\left(\ell^2-\frac\right) \rightarrow \ell^2. This manipulation is also equivalent to insert a 1/4 constant factor whenever \ell^2 appears.


Justification

An even more convincing calculation is the derivation of
Regge trajectories Regge may refer to * Tullio Regge (1931-2014), Italian physicist, developer of Regge calculus and Regge theory * Regge calculus, formalism for producing simplicial approximations of spacetimes * Regge theory, study of the analytic properties of sc ...
(and hence eigenvalues) of the radial Schrödinger equation with
Yukawa potential In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form: :V_\text(r)= -g^2\frac, where is a m ...
by both a perturbation method (with the old \ell(\ell+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 book and for the WKB calculation Boukema.


See also

*
Einstein–Brillouin–Keller method The Einstein–Brillouin–Keller method (EBK) is a semiclassical method (named after Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from Bohr- ...


References

{{DEFAULTSORT:Langer Correction Theoretical physics