Time development of a wave packet and the time delay

A one-dimensional scattering problem off a $\delta$-shaped potential is solved analytically and the time development of a wave packet is derived from the time-dependent Schr\"odinger equation. The exact and explicit expression of the scattered wave packet supplies us with interesting information about the "time delay" by potential scattering in the asymptotic region. It is demonstrated that a wave packet scattered by a spin-flipping potential can give us quite a different value for the delay times from that obtained without spin-degrees of freedom.Comment: 13 pages, plain TeX, 2 eps figures, tar+gzip+uuencode

Multiphoton Ionization as Time-Dependent Tunneling

A new semiclassical approach to ionization by an oscillating field is presented. For a delta-function atom, an asymptotic analysis is performed with respect to a quantity h, defined as the ratio of photon energy to ponderomotive energy. This h appears formally equivalent to Planck's constant in a suitably transformed Schroedinger equation and allows semiclassical methods to be applicable. Systematically, a picture of tunneling wave packets in complex time is developped, which by interference account for the typical ponderomotive features of ionization curves. These analytical results are then compared to numerical simulations and are shown to be in good agreement.Comment: 36 pages (also printable half size), uuencoded compressed tarred Latex file with 9 Postscript figures included automaticall

Second-price auctions with private entry costs

We study asymmetric second-price auctions under incomplete information. The bidders have potentially different, commonly-known, valuations for the object and private information about their entry costs. The seller, however, does not benefit from these entry costs. We calculate the equilibrium strategies of the bidders and analyze the optimal design for the seller in this environment

Quantum Tunneling in the Wigner Representation

Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one - dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semi-classical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles transmitted through or reflected from the potential barrier, as well as for elucidating the tunneling time problem.Comment: 18 pages, LATEX, no figure