Langer correction explained

The Langer correction, named after the mathematician Rudolf Ernest Langer, 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,$$-\backslash frac\; \backslash frac\; +\; [E-V\_\backslash textrm\{eff\}(r)]\; R(r)\; =\; 0,$$where the effective potential is given by$$V\_\backslash textrm(r)\; =\; V(r)\; -\; \backslash frac$$(

\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.

In 1937, Rudolf E. Langer suggested a correction$$\backslash ell(\backslash ell+1)\; \backslash rightarrow\; \backslash left(\backslash ell+\backslash frac\backslash right)^2$$which is known as Langer correction or Langer replacement.^[1] 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.^[2]

In 2D systems

Note that for 2D systems, as the effective potential takes the form$$V\_\backslash textrm(r)\; =\; V(r)\; -\; \backslash frac,$$so Langer correction goes:^[3] $$\backslash left(\backslash ell^2-\backslash frac\backslash right)\; \backslash rightarrow\; \backslash ell^2.$$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 (and hence eigenvalues) of the radial Schrödinger equation with Yukawa potential 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^[4] and for the WKB calculation Boukema.^{[5] [6]}

See also

• Einstein–Brillouin–Keller method

Notes and References

1. Langer . Rudolph E. . On the Connection Formulas and the Solutions of the Wave Equation . Physical Review . American Physical Society (APS) . 51 . 8 . 1937-04-15 . 0031-899X . 10.1103/physrev.51.669 . 669–676. 1937PhRv...51..669L .
2. Harald J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. World Scientific (Singapore, 2012), p. 404.
3. Book: Brack. Matthias. Semiclassical Physics. Bhaduri. Rajat. 2018-03-05. CRC Press. 978-0-429-97137-2. 76.
4. Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012), Chapter 16.
5. Boukema. J.I.. 1964. Calculation of regge trajectories in potential theory by W.K.B. and variational techniques. Physica. Elsevier BV. 30. 7. 1320–1325. 10.1016/0031-8914(64)90084-9. 1964Phy....30.1320B. 0031-8914.
6. Boukema. J.I.. 1964. Note on the calculation of Regge trajectories in potential theory by the second-order W.K.B. approximation. Physica. Elsevier BV. 30. 10. 1909–1912. 10.1016/0031-8914(64)90072-2. 1964Phy....30.1909B. 0031-8914.