Position Measurement-Induced Collapse: A Unified Quantum Description of Fraunhofer and Fresnel Diffractions
Position measurement-induced collapse states are shown to provide a unified quantum description of diffraction of particles passing through a single slit. These states, which we here call `quantum location states', are represented by the conventional rectangular wave function at the initial moment of position measurement. We expand this state in terms of the position eigenstates, which in turn can be represented as a linear combination of energy eigenfunctions of the problem, using the closure property. The time-evolution of the location states in the case of free particles is shown to have position probability density patterns closely resembling diffraction patterns in the Fresnel region for small times and the same in Fraunhofer region for large times. Using the quantum trajectory representations in the de Broglie-Bohm, modified de Broglie-Bohm and Floyd-Faraggi-Matone formalisms, we show that Fresnel and Fraunhofer diffractions can be described using a single expression. We also discuss how to obtain the probability density of location states for the case of particles moving in a general potential, detected at some arbitrary point. In the case of the harmonic oscillator potential, we find that they have oscillatory properties similar to that of coherent states.