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Journal of Physics Conference Series

From waves to trajectories: some aspects of time in quantum problems

M Kleber

Published Feb 1, 2008• 0 citations
Publisher

Abstract

In stationary quantum mechanical problems time is absent. However time comes back if an experiment is performed on a stationary wave function. This property is demonstrated for a real experiment (photodetachment). The wave function is shown to set the stage for mutually exclusive experimental results, each of them having its own history of Feynman paths.

DOI
10.1088/1742-6596/99/1/012001

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In stationary quantum mechanical problems time is absent. However time comes back if an experiment is performed on a stationary wave function. This property is demonstrated for a real experiment (photodetachment). The wave function is shown to set the stage for mutually exclusive experimental results, each of them having its own history of Feynman paths.
Abstract Time plays a fundamental role in our ability to make sense of the physical laws in the world around us. The nature of time has puzzled people — from the ancient Greeks to the present day — resulting in a long running debate between philosophers and physicists alike to whether time needs change to exist (the so-called relatival theory), or whether time flows regardless of change (the so-called substantival theory). One way to decide between the two is to attempt to measure the flow of time with a stationary clock, since if time were substantival, the flow of time would manifest itself in the experiment. Alas, conventional wisdom suggests that in order for a clock to function, it cannot be a static object, thus rendering this experiment seemingly impossible. Here we show, with the aid of counterfactual measurements, the surprising result that a quantum clock can measure the passage of time even while being switched off, thus lending constructive support for the substantival theory of time.
Abstract Any microscopic object is assumed to be a point-like particle, fluctuating quite randomly in the time scale τ\tauτ, which is much smaller than the smallest time interval Δ\DeltaΔ we can discriminate. The trajectories in Δ\DeltaΔ scale are defined. They are exclusive of each other since they originate from one particle having a position which is an objectively existing value. The measurement of the object is performed by the wave function, which is defined by summing up trajectories in Δ\DeltaΔ scale together with the phase factor. The wave function is thus a linear combination of exclusive paths. The measurement is performed by the macrovariable denoted as X. It loses the fluctuation by the stationary phase mechanism. Using a simple model to measure the operator O of the object, it is shown that, when the detector is switched on, the stationary phase trajectory of X branches into many trajectories, each of which is mapped to the eigenstate of O. Although these trajectories are all summed up for the wave function, in the actual experiment X takes one stationary trajectory. Therefore, the object is frozen to the corresponding eigenstate and realizes the reduction. The process of the reduction is shown to be realized with the correct frequency when the experiment is repeated. A model of the position measurement is also studied and the estimation of the magnitude of the upper bound of Δ\DeltaΔ is given based on the present experimental data.
In quantum physics, disturbance due to a measurement is not negligible. This requires the time parameter ttt in the Schrödinger or Heisenberg equation to be considered differently from a time continuum of experimenter's clock TTT on which physical events are recorded. It will be shown that ttt represents an ensemble of time intervals on TTT during which a microsystem travels undisturbed. In particular t=0t=0t=0 represents the ensemble of preparation events that we refer to as the ensemble of beginnings of time. This restricts ttt to be $0\leq t
The authors show how complex paths can be consistently introduced into sums over Feynman histories by using the notion of functional contour integration. For a k-dimensional system specified by a potential with suitable analyticity properties, each coordinate axis is replaced by a copy of the complex plane, and at each instant of time a contour is chosen in each plane. This map from the time axis into the set of complex contours defines a functional contour. The family of contours labelled by time generates a (k+1)-dimensional submanifold of the (2k+1)-dimensional space defined by the cartesian product of the time axis and the coordinate planes. The complex Feynman paths lie on this submanifold. The convergence problems encountered in previous proposals for complex path integrals are avoided by the requirement that each contour is asymptotically pinched to the real coordinate axis. An application of this idea to systems described by absorptive potentials yields a simple derivation of the correct WKB result in terms of a complex path that extremalises the action. The method can also be applied to spherically symmetric potentials by using a partial wave expansion and restricting the contours appropriately.
A quantum mechanics representation based on position (r⃗\vec{r}r), linear momentum(p⃗\vec{p}p​) and energy(EEE) eigenvalues is presented here. A set of equations, explicitly independent on wave function, was derived relating these observables. In this view, a particle has a known trajectory and at any point on space there is a linear momentum associated. Trajectory here, can be viewed as if a measurement were taken continuously. This picture does not change current quantum mechanics interpretation, rather it comes as a new route of calculation. Also, wave function can be retrieved performing an integration in all space for linear momentum. Equations derived in this work, present a potential dependent on linear momentum originating what we call quantum force. A particle experiences a total force whose resultant is composed by classic and quantum force, evolving on time according to it. In terms of evaluation, a single particle could be described by a set of auxiliary particles named as wave particles(WP). They will start with a range of initial conditions and will collectively describe probabilistic aspects of quantum mechanics. We expect this route could be applied to many problems, such as multi-body systems.
In quantum mechanics, time plays a role unlike any other observable. We find that measuring whether an event happened, and measuring when an event happened are fundamentally different - the two measurements do not correspond to compatible observables and interfere with each other. We also propose a basic limitation on measurements of the arrival time of a free particle given by 1/Ek where Ek is the particle's kinetic energy. The temporal order of events is also an ambiguous concept in quantum mechanics. It is not always possible to determine whether one event lies in the future or past of another event. One cannot measure whether one particle arrives to a particular location before or after another particle if they arrive within a time of 1/E of each other, where E is the total kinetic energy of the two particles. These new inaccuracy limitations are dynamical in nature, and fundamentally different from the Heisenberg uncertainty relations. They refer to individual measurements of a single quantity. It is hoped that by understanding the role of time in quantum mechanics, we may gain new insight into the role of time in a quantum theory of gravity.