Position Measurement-Induced Collapse: A Unified Quantum Description of Fraunhofer and Fresnel Diffractions

Abstract

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 timeevolution 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. * moncyjohn@yahoo.co.uk † kiran007x@yahoo.co.in PMIC: A Unified Quantum Description of Fraunhofer and Fresnel Diffractions 2

Position measurement-induced collapse states: Proposal of an experiment

“…By this approach, only the Fraunhofer-type patterns were obtained on the screen, for all distances from the slit. In contrast, in [7] we postulated that the time-evolution of the collapsed state is described by the PMIC wave function. It was shown that this results in a single expression that describes the Fresnel and Fraunhofer diffraction.…”

“…In this paper we have also used the trajectory representation in quantum mechanics, which accept the existence of particle trajectories associated with the wave functions. In [7], we have used this additional postulate to obtain the single expression that gives a unified description of Fresnel and Fraunhofer diffractions. It was noted that all the three trajectory formulations make identical predictions in this case and that the success of our PMIC description supports the existence of particle trajectories.…”

“…In [7], the PMIC states is defined using this expression for collapsed states. First assume that immediately after the above collapse, the particle is in a potential V , where the eigenstates of energy are |u n and the corresponding position space energy eigenfunctions are u n (y) ≡ y|u n .…”

“…The probability distribution corresponding to this total wave function is independent of x and z; it depends only on y and t. The problem of obtaining a stationary diffraction pattern on the screen can be solved if we assume trajectories along the x-direction, from the slit to the screen, as was done in [7]. The postulate of existence of particle trajectories, in addition to the standard postulates of quantum mechanics, is characteristic of nonlocal hidden variable theories such as the de Broglie-Bohm (dBB) [11], modified de Broglie-Bohm (MdBB) [12], and the Floyd-Faraggi-Matone (FFM) [13,14,15] trajectory formulations.…”

“…The question whether single-slit diffraction can be treated as quantum measurement of position of a particle was raised by Marcella [4] and was followed up in [5,6]. In a recent paper, we have obtained a satisfactory, affirmative answer [7] to the above question using the standard axioms of quantum mechanics, together with nonlocal quantum trajectory representations. As in [4], we have considered a particle that passes through a slit at a time t = t M and assumed that its wave function collapses to a rectangular function.…”