Langer correction
The Langer correction, named after the mathematician Rudolf Ernest Langer, is a correction to the WKB approximation for problems with radial symmetry.
Description[edit]
In 3D systems[edit]
When applying WKB approximation method to the radial Schrödinger equation,
In 1937, Rudolf E. Langer suggested a correction
In 2D systems[edit]
Note that for 2D systems, as the effective potential takes the form
Justification[edit]
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 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[edit]
References[edit]
- ^ Langer, Rudolph E. (1937-04-15). "On the Connection Formulas and the Solutions of the Wave Equation". Physical Review. 51 (8). American Physical Society (APS): 669–676. Bibcode:1937PhRv...51..669L. doi:10.1103/physrev.51.669. ISSN 0031-899X.
- ^ Harald J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. World Scientific (Singapore, 2012), p. 404.
- ^ Brack, Matthias; Bhaduri, Rajat (2018-03-05). Semiclassical Physics. CRC Press. p. 76. ISBN 978-0-429-97137-2.
- ^ Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012), Chapter 16.
- ^ Boukema, J.I. (1964). "Calculation of regge trajectories in potential theory by W.K.B. and variational techniques". Physica. 30 (7). Elsevier BV: 1320–1325. Bibcode:1964Phy....30.1320B. doi:10.1016/0031-8914(64)90084-9. ISSN 0031-8914.
- ^ Boukema, J.I. (1964). "Note on the calculation of Regge trajectories in potential theory by the second-order W.K.B. approximation". Physica. 30 (10). Elsevier BV: 1909–1912. Bibcode:1964Phy....30.1909B. doi:10.1016/0031-8914(64)90072-2. ISSN 0031-8914.