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arXiv:1905.13023 · Lithuanian Journal of Physics

Classical analog to the Airy wave packet

Algirdas Matulis, A. Acus

Published Oct 28, 2019• 1 citation
arXivHTMLPDF

Abstract

The solution of the Liouville equation for the ensemble of free particles is presented and the classical analog to the quantum accelerating Airy wave packet is constructed and discussed. Considering the motion of various classical packets – with an infinite and restricted distribution of velocities of particles – and also the motion of their fronts, we demonstrate in the simplest and most definite way why the packet can display a more sophisticated behaviour (even acceleration) as compared with a free individual particle that moves at a fixed velocity. A comparison of this classical solution with the quantum one in the Wigner representation of quantum mechanics, which provides the closest analogy, is also presented.

DOI
10.3952/physics.v59i3.4078

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The solution of the Liouville equation for the ensemble of free particles is presented and the classical analog to the quantum accelerating Airy wave packet is constructed and discussed. Considering the motion of various classical packets – with an infinite and restricted distribution of velocities of particles – and also the motion of their fronts, we demonstrate in the simplest and most definite way why the packet can display a more sophisticated behaviour (even acceleration) as compared with a free individual particle that moves at a fixed velocity. A comparison of this classical solution with the quantum one in the Wigner representation of quantum mechanics, which provides the closest analogy, is also presented.
In 1979, Berry and Balazs [M V Berry and N L Balazs 1979 Am. J. Phys. 47 264] obtained a strict solution of the Schrödinger equation with Airy function used as the initial condition, and described the wave function represented by such solution as the Airy wave-packets. They discovered that infinite Airy wave-packet has unique properties such as non-spreading and free acceleration, proving that it is the only nontrivial non-spreading solution of the time-dependent Schrödinger equation in one dimension. However, the observing of the finite Airy beam seems to be more meaningful since wave-packets in reality is inevitably band limited. A certain form of finite Airy beam was investigated by Siviloglou et al. in 2007 [Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901; Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979]. They noted that the Airy wave packet still exhibits its most exotic feature, i.e., its trend toward free acceleration. While in the present paper we discuss the properties of Airy beam in a few steps further and propose several conclusions. On the one hand, a theoretic explanation is given to solve the matter of the centre of mass of infinite Airy beam. On the other hand, deeper research is conducted on the unique properties of finite Airy beam. Another form of finite Airy beam is discussed by reduction to absurdity, and its field distribution is put forward by numerical simulation. We find that the trajectory of the centroid holds its position, which means that the beam cannot accelerate freely as a whole. Ultimately, we have the conclusion that finite Airy beam can neither freely accelerate nor be non-diffractive.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. 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On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. 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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.
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.
We simulate the center of mass motion of cold atoms in a standing, amplitude modulated, laser field as an example of a system that has a classical mixed phase-space. We show a simple model to explain the momentum distribution of the atoms taken after any distinct number of modulation cycles. The peaks corresponding to a classical resonance move towards smaller velocities in comparison to the velocities of the classical resonances. We explain this by showing that, for a wave packet on the classical resonances, we can replace the complicated dynamics in the quantum Liouville equation in phase-space by the classical dynamics in a modified potential. Therefore we can describe the quantum mechanical motion of a wave packet on a classical resonance by a purely classical motion.