Transient effects in two channel interactions and an application to the behavior of a time dependent shutter

Abstract

The subject of transient effects in quantum mechanics has been of interest to one of the authors (MM) since long before it became possible to study them experimentally. In particular, the problem of opening a shutter instantaneously led him to the concept of diffraction in time\cite{1}. Physically, it is only possible to open a shutter as a function of time and this complicates the problem greatly, as it is then not invariant under time translations and thus, among other difficulties, the energy of the system is not a constant of the motion. Kleber and Scheitler\cite{2} analyzed the problem describing the shutter as a $\delta-$function potential at the origin $x=0$, but whose strength was an inverse function of time. In this paper, we follow another procedure by adding to our initial two particle channel another one, and make them interact through appropriate time independent boundary conditions at the point of coincidence. The full problem conserves the total energy, but this does not happen if we restrict ourselves to a description in the first channel only. Thus, we have a rough but analytically solvable problem using a Laplace transform, which mimics some aspects of a time-dependent shutter, and compare our results with some of those derived by the different procedure of reference\cite{2}.