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arXiv:1702.07683 · Journal of Physics B Atomic Molecular and Optical Physics

Classical hallmarks of macroscopic quantum wave function propagation

James M. Feagin, J S Briggs

Published Jul 11, 2017• 7 citations
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Abstract

The precise connection between quantum wave functions and the underlying classical trajectories often is presented rather vaguely by practitioners of quantum mechanics. Here we demonstrate, with simple examples, that the imaging theorem (IT) based on the semiclassical propagator provides a precise connection. Wave functions are preserved out to macroscopic distances but the variables, position and momentum of these functions describe classical trajectories. We show that the IT, based on an overtly time-dependent picture, provides a strategy alternative to standard scattering theory with which to compare experimental results to theory.

DOI
10.1088/1361-6455/aa7f01

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The precise connection between quantum wave functions and the underlying classical trajectories often is presented rather vaguely by practitioners of quantum mechanics. Here we demonstrate, with simple examples, that the imaging theorem (IT) based on the semiclassical propagator provides a precise connection. Wave functions are preserved out to macroscopic distances but the variables, position and momentum of these functions describe classical trajectories. We show that the IT, based on an overtly time-dependent picture, provides a strategy alternative to standard scattering theory with which to compare experimental results to theory.
Unbound wave packets propagating to macroscopic space and time coordinates become proportional to their (Fourier transform) momentum distribution at earlier times whereby the asymptotic coordinates and the initial momenta are connected everywhere by appropriate classical trajectories. This asymptotic imaging theorem is relevant to every quantum reaction involving macroscopic extraction and detection of fragments emerging from a microscopic volume. It justifies the usual assumption of classical particle motion used in the design of particle detectors. We illustrate this quantum to semiclassical transition with the example of the free propagation of a wave packet in one dimension, a standard problem treated in introductory lectures on quantum mechanics. We indicate generalizations appropriate for more advanced discussions.
Conventional scattering theory is incomplete in that it does not adequately describe the behaviour of the wave function at macroscopic distances from the scattering reaction volume. In scattering experiments particles are incident from sources at macroscopic distance and measured at macroscopic distance from the microscopic reaction volume. Standard experimental procedure is to describe asymptotic particle motion by classical mechanics. However, this transition to classical mechanics is not accounted for in standard quantum scattering theory. Hence, we revisit conventional scattering theory with the introduction of semiclassical propagators to describe the wave function in both incident and outgoing asymptotic scattering channels. This description leads to the manifestation of classical trajectories in the quantum wave function both before and after the collision. It includes the case of extraction with external fields as in modern multi-particle detectors. We derive the asymptotic Kemble imaging theorem (IT) limit to demonstrate how formal quantum scattering theory contains implicit classical mechanics threaded throughout the asymptotic wave functions.
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The mechanism of the transition of a dynamical system from quantum to classical mechanics is one of the remaining challenges of quantum theory. Currently, it is considered to occur via decoherence caused by entanglement and/or stochastic interaction with a quantum environment. Here we prove that, in the absence of de-phasing environmental interaction, the asymptotic spatial wave function of any quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to arbitrary deterministic fields, becomes proportional to the initial momentum wave function, \emph{where the particle positions and momenta are related by classical mechanics.} This implies that detection of particle positions and momenta at different times will yield results in accordance with classical mechanics, without the need to postulate decoherence of the wave function.
The mechanism of the transition of a dynamical system from quantum to classical mechanics is of continuing interest. Practically it is of importance for the interpretation of multi-particle coincidence measurements performed at macroscopic distances from a microscopic reaction zone. Here we prove the generalized iimaging theorem which shows that the spatial wave function of any multi-particle quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to deterministic external fields and particle interactions, becomes proportional to the initial momentum wave function where the position and momentum coordinates define a classical trajectory. Currently, the quantum to classical transition is considered to occur via decoherence caused by stochastic interaction with an environment. The imaging theorem arises from unitary Schroedinger propagation and so is valid without any environmental interaction. It implies that a simultaneous measurement of both position and momentum will define a unique classical trajectory, whereas a less complete measurement of say position alone can lead to quantum interference effects.
Abstract The free propagation in time of a normalisable wave packet is the oldest problem of continuum quantum mechanics. Its motion from microscopic to macroscopic distance is the way in which most quantum systems are detected experimentally. Although much studied and analysed since 1927 and presented in many textbooks, here the problem is re‐appraised from the standpoint of semi‐classical mechanics. Particular aspects are the emergence of deterministic trajectories of particles emanating from a region of atomic dimension and the interpretation of the wave function as describing a single particle or an ensemble of identical particles. Of possible wave packets, that of Gaussian form is most studied due to the simple exact form of the time‐dependent solution in real and in momentum space. Furthermore, this form is important in laser optics. Here the equivalence of the time‐dependent Schrödinger equation to the paraxial equation for the propagation of light is demonstrated explicitly. This parallel helps to understand the relevance of trajectory concepts and the conditions necessary for the perception of quantum motion as classical. Key Points The paper addresses a problem of fundamental quantum mechanics, important in both physics and chemistry. Within physics, the subject is relevant to the disciplines of both quantum mechanics and classical optics. The matter presented is topical also for researchers in the history and philosophy of physics.