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The European Physical Journal Special Topics

Mean square displacement of a free quantum particle on the basis of thermal Gaussian wave packets

Oussama Bindech, Roberto R. Marquardt

Published Aug 7, 2023• 3 citations
Publisher
DOI
10.1140/epjs/s11734-023-00920-7

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The mean square displacement x(t)−x(0)2 of the position x of a free particle of mass m at thermal equilibrium is evaluated quantum mechanically. An analytical expression is obtained which shows an initial quadratic increase of the mean square displacement with time and later on a linear growth, with the slope ℏ/m, quite at variance with the result from classical statistical mechanics. Results are discussed in relation to observables from helium scattering or spin-echo experiments, and their possible interpretation in terms of the classical and quantum mechanical expression for the mean square displacement of an essentially free particle.
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 specific features of scattering of the Gaussian wave packet by the simplest of all reflectionless potentials, the Sech-squared well, are considered. The results of numerical computation show that a wave packet that has passed through this potential overtakes a free wave packet and is both higher and narrower than a free wave packet; in other words, the scattered wave packet accelerates inside the well and preserves its shape better. Comparisons with scattering by the square well and by a series of reflectionless potentials are also presented. Possible applications of the Sech-squared as well as other reflectionless potentials are discussed.