The behaviour of localized quantum wave packet has lately become an object of much interest to forge a link between classical and quantum mechanics. Wave packet is often taken as the best approximation of quantum theory to a classical particle. If the wave packet is regarded as a point, with zero amplitude everywhere except in a very small region, then its motion can be interpreted as the motion of a single particle. The advent of short-pulsed lasers made it possible to produce and detect superposition of electron states for many physical systems. Such superposition leads to the formation of localized electron wave packets. The localized wave packet solution disperses or spreads in time, however, at a later time, it reforms and appears as its initial state. Revivals in quantum systems were first studied in the Jaynes–Cumming model, describing a two-level atom interacting with a resonant monochromatic field.
The behaviour of localized quantum wave packet has lately become an object of much interest to forge a link between classical and quantum mechanics. Wave packet is often taken as the best approximation of quantum theory to a classical particle. If the wave packet is regarded as a point, with zero amplitude everywhere except in a very small region, then its motion can be interpreted as the motion of a single particle. The advent of short-pulsed lasers made it possible to produce and detect superposition of electron states for many physical systems. Such superposition leads to the formation of localized electron wave packets. The localized wave packet solution disperses or spreads in time, however, at a later time, it reforms and appears as its initial state. Revivals in quantum systems were first studied in the Jaynes–Cumming model, describing a two-level atom interacting with a resonant monochromatic field.
A consistent description of the motion of a particle by wave propagation implies not only a geometrical equivalence between a trajectory and a ray but also a kinematical equivalence between the time rates at which the particle or wave is traversing its path. This may be tentatively achieved by using a wave configuration, known as a wave packet, represented as a continuous superposition of monochromatic waves. If the wave packet can be regarded as a point, having zero amplitude everywhere except in a very small region, its motion along a trajectory might be interpreted as the motion of a single particle. The representation of a particle in space and time by means of a highly localized wave packet is a basic assumption of wave mechanics. The behavior of the wave packet as a whole will account for experiments where particle-like properties are observed, whereas individual wavelength components of packet will appear in experiments which depend upon interference and diffraction effects.
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.
An attempt to explain the motion of particle is made in terms of a wave superposition. There is no localized superposition of waves that is supposed to be a single particle apart from other wave superposition, but there are a bunch of wave packets, which appears to be particle within a wave superposition that is made of individual waves around the space. There are countless number of waves that become a wave superposition, and the position of each wave packet is determined by the relative distance between packets, which is a superposition state. When a wave superposition interfere with a individual wave, this distance changes, and the position of each wave packet is fixed , which is called a measurement. With respect to the position of a particle, thus the particle is placed at the different position simultaneously in the superposition state. A change of the relative distance between wave packets appears to be a motion of particle, which is interpreted as a force acting on a particle. The photo-electric effect is explained in the same way, by which a superposition of charge waves interferes with the light wave, which increases the relative distance of wave packets in the atom, and it brings the outermost wave packet apart from the atom. The resistance in conductor is also understood by the wave interference with a superposition of charge waves residing in the conductor.
The wave-particle duality introduced by de Broglie and developed mathematically by Schrödinger remains one of the deepest conceptual achievements of modern physics. Yet Schrödinger's original physical intuition contained a problem that has never entirely disappeared: if a particle is represented by a wave, how can an extended and generally dispersive wave structure correspond to the localized events observed in experiments? Schrödinger initially hoped that a particle could be understood as a localized wave packet formed by the coherent superposition of many elementary waves. The program was attractive because localization could then arise from interference rather than being imposed as an independent particle property. However, generic wave packets disperse under free evolution, while stable bound states of quantum systems are not generally localized packets moving along classical trajectories. The subsequent Born interpretation provided an extraordinarily successful connection between the wave function and experimental statistics, identifying ∣Ψ∣2|\Psi|^2 with a probability density. Nevertheless, this interpretation leaves open a distinct physical question: how does a spatially extended complex amplitude structure give rise to localized, discrete detection events? This paper revisits that question from the historical development of wave mechanics to the operational structure of early scattering experiments. Particular attention is given to the experiments of Davisson and Kunsman, Ramsauer's measurements of slow-electron scattering, and Dymond's measurements of the velocity and angular distribution of electrons scattered in helium. Although these experiments differ in physical arrangement and measured quantities, they share a common operational structure: an extended incident quantum state undergoes propagation and interaction, after which the experiment registers discrete events within a finite spatial, angular, or energy acceptance region. The central proposal of this paper is therefore not a new derivation of Born's rule, but a reformulation of the problem that precedes such a derivation. Instead of asking only why a quantum state has a probability distribution, one may ask how a propagating wave-amplitude structure is dynamically converted into a localized event rate. In symbolic form, Ψ⟶D⟶Γ,\Psi \quad \longrightarrow \quad \mathcal{D} \quad \longrightarrow \quad \Gamma, where Ψ\Psi denotes the propagating amplitude structure, D\mathcal{D} represents the interaction and detection dynamics, and Γ\Gamma denotes the resulting localized event rate. Born's rule then appears as the empirical relation Γ(r)∝∣Ψ(r)∣2,\Gamma(\mathbf r)\propto |\Psi(\mathbf r)|^2, whose dynamical origin remains to be established. The paper concludes by reconsidering Schrödinger's original wave intuition rather than simply declaring it defeated. The failure of the persistent localized wave packet may indicate not that wave ontology was fundamentally misguided, but that propagation and localization are physically distinct stages of quantum dynamics. This distinction provides a conceptual starting point for a subsequent series of investigations into the dynamical origin of quantum probability.
In quantum mechanics the situation is different because the uncertainty principle does not allow us to know the initial positions and momenta of the particles with desired accuracy. The individual particle is described by a wave packet. Since the wave packet is localized and spreading with time, it is impossible to distinguish two identical particles as their wave functions always overlap at any point in space. Quantum theory describes any object at energies below its excitation threshold with four degrees of freedom. The wave functions of all the states of a given system of identical particles must have the same symmetry; otherwise, the wave function of a state formally a superposition of states of different symmetry would be neither symmetrical nor antisymmetrical. Hence, for a system of identical particles, at high energies and low concentration, probable number of particles per quantum state will be very small. Wave functions of the particles are well separated.