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.