In this paper, wave propagation in one-dimensional few- and many-particle systems, all formed by interacting identical particles, is discussed for the purpose of explaining why wave propagation always has a finite speed in many-particle systems. After studying the equation of motion of each particle, it is demonstrated that such a finite speed is an observable speed realized after instantaneous wave propagation in few-particle systems is suppressed in many-particle systems by the large particle numbers in the latter systems. The expression of the speed is obtained and studied for its properties.
In this paper, wave propagation in one-dimensional few- and many-particle systems, all formed by interacting identical particles, is discussed for the purpose of explaining why wave propagation always has a finite speed in many-particle systems. After studying the equation of motion of each particle, it is demonstrated that such a finite speed is an observable speed realized after instantaneous wave propagation in few-particle systems is suppressed in many-particle systems by the large particle numbers in the latter systems. The expression of the speed is obtained and studied for its properties.
Introduction In this chapter we discuss the basic physical concepts and equations related to the behavior of particles in the nanoworld. We introduce the Schrödinger wave equation for particles and determine the ways to calculate observable physical quantities. We find that, in wave mechanics, the motion of a particle confined to a finite volume is always characterized by discrete values of the energy and standing-wave-like wavefunctions, i.e., such a motion is quantized. While motion in an infinite space (i.e., free motion) is not quantized and is described by propagating waves, the energy of the particle is characterized by a continuous range of values. Keeping in mind the diverse variants of nanostructures, by using wave mechanics we analyze some particular examples, which highlight important quantum properties of particles. Many of the examples analyzed can serve as the simplest models of nanostructures and will be exploited in following chapters to understand the fundamentals of processes in nanoelectronics. The Schrödinger wave equation From the previous chapter, we conclude that nanosize physical systems are quantum-mechanical systems, inasmuch as their sizes are comparable to typical de Broglie wavelengths of the particles composing these systems. In dealing with quantum-mechanical systems, one aims at determining the wavefunction of a single particle or of the whole system. As we will demonstrate in the subsequent discussion, knowledge of the wavefunction in quantum mechanics is sufficient to describe completely a particle or even a system of particles.
On physical grounds, one expects locally interacting quantum many-body systems to feature a finite group velocity. This intuition is rigorously underpinned by Lieb-Robinson bounds that state that locally interacting Hamiltonians with finite-dimensional constituents on suitably regular lattices always exhibit such a finite group velocity. This also implies that causality is always respected by the dynamics of quantum lattice models. It had been a long-standing open question whether interacting bosonic systems also feature finite speeds of sound in information and particle propagation, which was only recently resolved. This work proves a strikingly simple such bound for particle propagation - shown in literally a few elementary, yet not straightforward, lines - for generalized Bose-Hubbard models defined on general lattices, proving that appropriately locally perturbed stationary states feature a finite speed of sound in particle numbers.
Discrete wave mechanics is formulated for particles in one-dimensional systems by use of a simple finite difference equation. The solutions involve wave vectors (instead of wave functions) as well as a newly defined "wave vector energy." In the limit, as c --> infinity, the treatment reduces to that of Schrödinger's wave mechanics. Specific calculations are made for completely free particles as well as for particles confined to a one-dimensional box. The results exhibit a striking compatibility with relativistic considerations. The wave vectors show properties that can be identified with particles and anti-particles-each possess identical probability distributions with energies that add up to zero.
The wave-particle duality of quantum mechanics has always been an unsolved problem in physics. This article attempts to use one of the properties to explain the other, so as to eliminate the confusion of quantum mechanics probability waves. Specifically, this article finds that the discreteness of energy is the inherent property of all waves, so this article explains the particle nature of light from the perspective of light wave, thereby eliminating the confusion of light’s wave-particle duality; in addition, this article found that microscopic matter particles are only suitable for discussing the number of scattered particles or energy flow density, not their position and momentum, when they are forced to discuss their position and momentum, it will inevitably lead to confusion about probability waves, when only discussing the number of scattered particles or energy flow density, its volatility can be explained from the perspective of particle nature, thereby eliminating the confusion of microscopic matter particle probability waves. When the attribute of light as a wave is established, the light needs to overcome the Hamiltonian of the medium in different constraint systems (gravitational fields) during the propagation process, which will produce a universal redshift phenomenon, this can provide a new understanding of cosmic redshift; when the property of light as a wave is established, it means that the speed of light is a constant speed relative to the medium, which can provide a new understanding of the principle of constant speed of light.
Although many of the principles of quantum mechanics can be adequately illustrated by considering systems that consist of only one particle subject to external forces, there are a number of important phenomena that are manifest only in systems containing two or more particles, and these will be the subject of the present chapter. We shall begin with some general statements and go on to consider the case of two interacting particles subject to no external forces, when we shall find that, in this case, the problem can be separated into one describing the behaviour of the centre of mass of the system and another describing the relative motion. We shall then consider the case of two noninteracting particles and show that these can often be treated independently as would be expected. In the case of indistinguishable particles, however, we shall find that a symmetry is imposed on the wave function that ensures that the behaviour of such particles is coupled even when they do not interact. We shall introduce the problem of interacting systems containing more than two identical particles, explaining what is meant by “fermions” and “bosons” and how their properties compare. These results will be illustrated by discussing the energy levels and optical spectra of the helium atom, starting with the approximation that the two electrons do not interact and extending the discussion to the realistic case using perturbation theory. The chapter closes with a consideration of the effects of particle indistinguishability on scattering theory.
The "mechanics of a system of particles is developed on the basis of Schr\"odinger's wave-equation," without any use of matrices; it is shown that simple cases can be handled as easily by this method as by the matrix method. The passage to classical theory as an approximation stands out very clearly.The measurement of velocity, momentum, energy and angular momentum as secondary physical qualities is discussed, the usual probability amplitudes being obtained.The conservation of momentum and of energy is treated in terms of the new theory.