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arXiv:1709.05030

Virtual Detector Theory of Free-Particle Wave Packets

Xu Wang

Published Sep 15, 2017• 0 citations
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Abstract

We apply a recently emerging "virtual detector" method to general free-particle wave packets. We discuss what is actually detected by a virtual detector and give analytical expressions in both the real space and the momentum space. We prove that in the asymptotic limit when the time difference between wave packet creation and detection is large, resembling real detection processes, the obtained differential momentum distribution is equivalent to Fourier transforming the free-particle wave packets. We further consider the situation with the presence of laser fields.

DOI
10.48550/arxiv.1709.05030

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We apply a recently emerging "virtual detector" method to general free-particle wave packets. We discuss what is actually detected by a virtual detector and give analytical expressions in both the real space and the momentum space. We prove that in the asymptotic limit when the time difference between wave packet creation and detection is large, resembling real detection processes, the obtained differential momentum distribution is equivalent to Fourier transforming the free-particle wave packets. We further consider the situation with the presence of laser fields.
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.
The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.
We discuss four general features of the force-free evolution of wave packets: (1) The spatial spread of a packet changes with time in a simple way. (2) For sufficiently short durations (related to the spread in the momentum of the packet) the probability distribution will move with uniform speed and little change in shape. (3) After a sufficiently long time (related to the initial spatial spread) the wave function converges to a simple form that is simply related to the momentum distribution of the packet. In this asymptotic regime the shape of the probability distribution no longer changes, and its scale increases linearly with the time. (4) There is an infinite denumerable set of simple wave packets (the Hermite-Gauss packets) that do not change shape at any time during their evolution.
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 text books, 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 dimensions 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 Schroedinger 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 motion as classical.
The physics of special relativity is applied to a wave-packet model of a particle without the usual de Broglie-like assumption of Planck's rule, E = hv. The velocity of the particle is equated to the group-velocity of the wave-packet to associate the particle with its wave-packet. With such an identification it is demonstrated using special relativity that the momentum four-vector, P = [p, E/c2] of the particle is parallel and proportional to a central wave-number four-vector, K0 = [k0. ω0/c2], of the particle's wave-packet. The constant of proportionality, relating these two four-vectors, has the dimension of action and by diffraction experiments it has the value h/2π where h is Planck's constant. This refinement of the de Broglie wave-like nature of a particle suggests that special relativity underlies quantum physics more deeply than heretofore believed. The above quantum rules are used to find the wave-packet function in the one-dimensional case of a non-relativistic free particle and a particle in a conservative force field of potential V (x). This wave-packet function is shown to satisfy an integral time-evolution equation which relates the wave function at time t0 to the function at a slightly later time t. The resulting integral equation is equivalent to the propagator equation or path-integral hypothesised by Feynman. This fact also demonstrates, using the method of Feynman, that the wave-packet function satisfies Schrödinger's partial differential equation. The wave-packet function of a particle in a force field is directly shown to satisfy a linear integral equation which relates the wave-packet at one instant of time to its values at all later times. The kernel or propagator of this integral equation is also equivalent to Feynman's path integral. A computation demonstrates that the position of a moving particle in a simple quadratic potential well does not become more uncertain with the passage of time. This contrasts with the behaviour of a moving free particle which as a function of time becomes more and more difficult to accurately locate.
The particle-interference-experiments, with the photons and electrons have clearly shown that they are always detected as localized ‘particles’; and the density of their detections exhibit a pattern similar to the interference of waves. This experimental-observation is alternatively interpreted here, different from the current, ‘probabilistic interpretation’. The ‘particle’ is first mathematically characterized as an ‘impulse function’ in space; and then Fourier-transformed into the wave-number-domain, as a wide band of constituent-waves as shown in the fig. 1&2. The ‘particle’ is an ‘event’ of constructive-superimposition of a wide band of ‘pre-quantum-waves’. This ‘event’ occurs in space at regular distances; and at regular instances in time. In the particle-interference-experiments, they are the wide band of ‘pre-quantum-waves which interfere physically. This physical interference of the pre-quantum-waves determines the position and time of emergence of the ‘impulse in space’ detectable as the ‘particle’. In the real-world-situation, the ‘pre-quantum-waves of a wide bandwidth arrive from all the directions, so the amplitude and phase of every constituent wave keep changing; so the place of constructive superimposition of all the spectral-components, keep varying unpredictably; and only the ‘probability’ of detection in a given volume is possible in practice. Thus, form the theoretical view-point, there are only the pre-quantum-waves; and ‘particle’ is a specific situation of constructive-superimposition of all the spectral-components of pre-quantum-waves; so there is no wave-particle-duality. The ‘probabilistic-interpretation’ is only a ‘practical-limitation’, detector being able to detect only the ‘particle’, and not the spectrum of constituent waves.