Diffraction in time with dissipation

Abstract

The more usual problems discussed in classical mechanics in elementary textbooks are the non-dissipative ones in which there is a Hamiltonian representing the energy as a constant of motion. The translation of this type of problem to quantum mechanics is very well known. Conversely, there are very simple classical mechanical problems that involve dissipation, but whose translation to quantum mechanics on the basis of a corresponding Hamiltonian is sometimes misinterpreted, since the underlying classical formalism involves non-canonical transformations that lead to non-unitary transformations of the quantum mechanical wavefunctions. In this paper we shall discuss the problem of a beam of particles of given momentum incident from the left on a shutter that is opened at time t = 0. The solution has the well known properties of diffraction in time, but we will analyse it quantum mechanically both when dissipation is and is not present. The objective is to get a better understanding of the effect of dissipation in the quantum mechanical picture and to estimate the influence of the above mentioned non-unitary transformations on this particular problem.