Uniform semiclassical approximations on a topologically non-trivial configuration space: The hydrogen atom in an electric field

Semiclassical periodic-orbit theory and closed-orbit theory represent a quantum spectrum as a superposition of contributions from individual classical orbits. Close to a bifurcation, these contributions diverge and have to be replaced with a uniform approximation. Its construction requires a normal form that provides a local description of the bifurcation scenario. Usually, the normal form is constructed in flat space. We present an example taken from the hydrogen atom in an electric field where the normal form must be chosen to be defined on a sphere instead of a Euclidean plane. In the example, the necessity to base the normal form on a topologically non-trivial configuration space reveals a subtle interplay between local and global aspects of the phase space structure. We show that a uniform approximation for a bifurcation scenario with non-trivial topology can be constructed using the established uniformization techniques. Semiclassical photo-absorption spectra of the hydrogen atom in an electric field are significantly improved when based on the extended uniform approximations

Fictitious time wave packet dynamics: I. Nondispersive wave packets in the quantum Coulomb problem

Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of "restricted Gaussian wave packets" consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically, and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well defined angular momentum quantum n umbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\v{c}i\v{c} et al., submitted].Comment: 12 pages, 3 figure

Closed orbits and their bifurcations in the crossed-fields hydrogen atom

A systematic study of closed classical orbits of the hydrogen atom in crossed electric and magnetic fields is presented. We develop a local bifurcation theory for closed orbits which is analogous to the well-known bifurcation theory for periodic orbits and allows identifying the generic closed-orbit bifurcations of codimension one. Several bifurcation scenarios are described in detail. They are shown to have as their constituents the generic codimension-one bifurcations, which combine into a rich variety of complicated scenarios. We propose heuristic criteria for a classification of closed orbits that can serve to systematize the complex set of orbits

What price civil justice?

In the jurisdictions both of England and Wales and of Scotland, the civil justice system is currently the subject of intense critical appraisal. This paper considers the current status of civil justice, beginning by asking what we expect from our system of civil justice and going on to analyse the supply and demand of civil legal services in market terms. The authors conclude that there is scope in the system for substantial experimentation with procedural reform. They suggest that as much as possible should be done to encourage the development of a cafeteria-style of civil justice system with improved information flows for consumers, and provision for innovations such as cost capping and risk sharing between legal representatives and clients, so providing consumers with more freedom of choice.

Pre-trial settlement: Who's for two-way offers?

This paper reviews the use of cost shifting devices intended to encourage pre-trial settlement. Both the well-known instrument of judicial offers (tenders) and the more recently introduced pursuers' offers are discussed. Numerical examples are provided and experimental evidence is reviewed. Both sources create some doubts regarding the efficacy of such devices in encouraging settlement. There is a strategic aspect of negotiating that is encouraged by these rules. In essence, these arrangements impart a certain amount of power to one side or other. This influences the level of settlement (if any), but may also reduce the probability of reaching a settlement. Abandoning such arrangements, while somewhat contrary to conventional views may well be a positive step in encouraging pre-trial settlement.