Irreversible Behaviour and Collapse of Wave Packets in a Quantum System with Point Interactions

Abstract

A system of a particle and a harmonic oscillator, which have pure point spectrum if uncoupled, is known to acquire absolutely continuous spectrum when the particle and the oscillator are coupled by a sufficiently strong point interaction. Here the simple dynamical mechanism underlying this phenomenon is exposed. The energy of the oscillator is proven to exponentially diverge in time, while the spatial probability distribution of the particle collapses into a delta function in the interaction point. On account of this result, a generalized model with many oscillators which interact with the particle at different points is argued to provide a formal model for approximate measurement of position, and collapse of wave packets.

New version: an error in a proof in the original version has been corrected with no prejudice for the conclusions. New results about generalized functions have been added. The text has been expanded . To appear in J.Phys. A Math. Gen

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