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Free-Particle Quantum Dynamics from a Single Physical History: A Single-History Record Framework Recovery of the Propagator, Schrödinger Evolution, and Gaussian Wave-Packet Spreading

Tong Wang

Published Sep 15, 2026• 0 citations
OpenAlex

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Trajectories and the perception of classical motion in the free propagation of wave packets

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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.

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The main principles of photon quantum mechanics describing its single-particle states by means of the wave function (wave packet) in the coordinate representation are given. The results of modeling of the propagation in space-time of wave packet with a Gaussian distribution of the momentum of a photon corresponding to femtosecond laser radiation are discussed. On the basis of the general ideas about the evolution of this packet and within the framework of constructed photon quantum mechanics, the quantum-mechanical explanation of the experimentally registered (previously) reduction of speed of the free photons, “structured” in the same way as the light in Bessel and Gaussian beams, is proposed. It is emphasized that in fact the photon is not a some formed quantum corpuscle, but it is a quasi-particle, which is the result of propagation of spin wave in physical vacuum, the nature of which should be considered at the Planck distances simultaneously with structure of fundamental particles.

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Tomographic Description of a Quantum Wave Packet in an Accelerated Frame

Sergio M. De Nicola, Renato Fedele, Dušan Jovanović +2 more

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton's law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.

Entropy•May 19, 2021•3 citations
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.
The main principles of photon quantum mechanics describing its single-particle states by means of the wave function (wave packet) in the coordinate representation are given. The results of modeling of the propagation in space-time of wave packet with a Gaussian distribution of the momentum of a photon corresponding to femtosecond laser radiation are discussed. On the basis of the general ideas about the evolution of this packet and within the framework of constructed photon quantum mechanics, the quantum-mechanical explanation of the experimentally registered (previously) reduction of speed of the free photons, “structured” in the same way as the light in Bessel and Gaussian beams, is proposed. It is emphasized that in fact the photon is not a some formed quantum corpuscle, but it is a quasi-particle, which is the result of propagation of spin wave in physical vacuum, the nature of which should be considered at the Planck distances simultaneously with structure of fundamental particles.
The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton's law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.