Measurement as absorption of Feynman trajectories: Collapse of the wave function can be avoided

Abstract

We define a measuring device (detector) of the coordinate of quantum particle as an absorbing wall that cuts off the particle's wave function. The wave function in the presence of such detector vanishes on the detector. The trace the absorbed particles leave on the detector is identifies as the absorption current density on the detector. This density is calculated from the solution of Schrödinger's equation with a reflecting boundary at the detector. This current density is not the usual Schrödinger current density. We define the probability distribution of the time of arrival to a detector in terms of the absorption current density. We define coordinate measurement by an absorbing wall in terms of 4 postulates. We postulate, among others, that a quantum particle has a trajectory. In the resulting theory the quantum mechanical collapse of the wave function is replaced with the usual collapse of the probability distribution after observation. Two examples are presented, that of the slit experiment and the slit experiment with absorbing boundaries to measure time of arrival. A calculation is given of the two dimensional probability density function of a free particle from the measurement of the absorption current on two planes.

Arrival time from the general theory of quantum time distributions

“…The arrival time problem has been attacked in the literature with many different approaches (see e.g. [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] or the review [22] of older approaches), but neither of these approaches (except [21]) seems to be a special case of a general systematic theory that can treat arrival time on an equal footing with all other phenomena, such as the decay time, in which time appears as a random variable. This, we believe, is a sufficient motivation to study in detail how arrival time can be described by the general systematic theory developed in [1].…”

“…An additional motivation for this work arises from a recent argument by Das and Struyve [23] that the theory [1] and other similar approaches [4,5] to arrival time based on an exponential formula of the form (1) are not adequate because they lead to physically unacceptable exponential time-damping. In this paper we show that, when the general theory developed in [1] is applied correctly, no exponential time-damping occurs.…”

Time dispersion in quantum electrodynamics

“…This problem is not a central focus of this work. In paper B, where this problem was a central focus, we took as a starting point an analysis by Marchewka and Schuss [63,65,64,66] who made a strong argument (from probability conservation) that the probability current correctly gives the detection rate.…”

Does the Heisenberg uncertainty principle apply along the time dimension?

“…A conceptually sounder approach is supplied by Marchewka and Schuss, who use Feynman path integrals to compute the time-of-arrival [29,30,31,32]. Unfortunately their approach has free parameters, which rules it out for use here.…”

“…Thus the two integrals are orthogonal and give rise to no interference." -Marchewka and Schuss [31] We turn to an approach from Marchewka and Schuss [29,30,31,32]. They use a path integral approach plus the assumption of an absorbing boundary.…”