Langer correction

Abstract: For two Coulombically interacting electrons in a quantum dot with harmonic confinement and a constant magnetic field, we show that time-dependent semiclassical calculations using the Herman?Kluk initial value representation of the propagator lead to eigenvalues of the same accuracy as WKB calculations with a Langer correction. The latter are restricted to integrable systems, however, whereas the time-dependent initial value approach allows for applications to high-dimensional, possibly chaotic dynamics and is extendable to arbitrary shapes of the potential.

Abstract: A new approach to the correction of the inaccuracies of Thomas–Fermi theory as applied to atoms is developed, in which the exact angular part of the energy eigenfunctions are accepted and the statistical treatment restricted to the radial motion. The Langer correction is used to represent the quantum depletion of electron density close to the nucleus. Orbital eigenvalues are assigned by a quantization scheme which produces exact energies for noninteracting electrons. Self‐interaction and spin polarization are allowed for. The predicted total energies are correct to about 4% for the first 20 atoms. Ionization energies and electron densities are also discussed.

Abstract: First, two conditions are specified for the lowest order Wentzel-Kramers- Brillouin quantization rule to yield exact results. These rules are related to the periodic orbit decomposition of the quantum density of states. This approach is then applied to supersymmetric quantum mechanics. It leads to a new derivation of the result that shape- invariant potentials give exact results when the classical action is calculated with the square of the super potential, but without the Maslov index or the Langer correction.

Abstract: An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius $r_0$, in the framework of Wentzel-Kramers-Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz and Grabert : The centrifugal term is expanded perturbatively (in powers of $\hbar$), decomposing it into 2 terms -- the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energies of the hydrogen atom, to the first order in $\hbar$, with all higher order corrections vanishing. In the present study, this formalism is extended to the case of radial potentials under hard wall confinement, to check whether the same argument holds good for such confined systems as well. As explicit examples, 3 widely known potentials are studied, which are of considerable importance in the theoretical treatment of various atomic phenomena involving atomic transitions, viz., the 3-dimensional harmonic oscillator, the hydrogen atom, and the Hulthen potential.