The Langer modification is an improvement in the WKB analysis of the radial Schrödinger equation. We derive a generalization of the Langer modification to any radial operator. For differential operators we write the modified classical symbols explicitly and show that the WKB wavefunctions with the modification have the exact limiting behavior for small radius. Unlike in the Schrödinger case, generally the modified radial analysis is not equivalent to the WKB analysis of the full problem before reduction by the spherical symmetry.

We present an experimental and theoretical study of a simple, passive system consisting of a birefringent, two-dimensional photonic crystal and a polarizer in series, and show that superluminal dispersive effects can arise even though no incident radiation is absorbed or reflected. We demonstrate that a vector formulation of the Kramers-Kronig dispersion relations facilitates an understanding of these counter-intuitive effects.Superluminal group velocities have been observed in a number of different physical systems. These include passive absorptive [1], passive reflective [2,3], and active transparent [4] media. There have also been numerous theoretical and experimental proposals to observe superluminality in the tunneling of electromagnetic wavepackets [5].Here we report the first experimental observation of superluminal effects in a passive system with neither absorption nor reflection. The effects arise because of a transfer of energy or interference between two modes of the electromagnetic field, in this case two different polarizations of light. We are able to interpret these results using a new vector formulation of the Kramers-Kronig relations.It is widely believed that the Kramers-Kronig (K-K) relations [6] require that passive systems be either absorptive or reflective in order to exhibit superluminal effects. In this paper, we demonstrate that this is not the case; while absorptive and reflective systems are required to have spectral regions of anomalous dispersion, the converse is not necessarily true. In fact, the exchange of energy between modes is a sufficient condition for superluminal propagation in any system. We show that these effects are consistent with causality.Our experimental system consists of a slab of highly birefringent two-dimensional (2D) photonic crystal and a linear polarizer, placed in series. The photonic crystal has fundamental and second-order photonic band gaps in the regions of 10 and 20 GHz, and displays strong birefringence with very high transmission in the frequency range between the two gaps [7]. The crystal itself is an 18-layer hexagonal array of hollow acrylic rods (outer diameter 1/2") with an air-filling fraction (AFF) of 0.60. The crystal was constructed using a method which we have previously described [8].We studied the transmission and dispersive properties of this system between the two band gaps, using an HP 8720A vector network analyzer (VNA). Microwaves were coupled to and from free space with polarization-sensitive horn antennae. The photonic crystal was placed in the far-field of the transmitter horn to ensure that planar wavefronts of a well-defined polarization were incident on it. The receiver horn was positioned immediately behind the photonic crystal on a direct line of sight with the transmitter horn. In addition, the crystal and receiver horn were placed inside a microwave-shielded box with an open square aperture, whose size was chosen to minimize diffraction effects while eliminating signal leakage around the crystal. This method has proven very...

Asymptotic expressions for Clebsch-Gordan coefficients are derived from an exact integral representation. Both the classically allowed and forbidden regions are analyzed. Higher-order approximations are calculated. These give, for example, six digit accuracy when the quantum numbers are in the hundreds.

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