CRC Press eBooks

Waves and Forces

Jules J. Berman

Published Dec 2, 2024• 0 citations

Abstract

This chapter describes how fundamental particles can be conceived as wave packets. In the course of this chapter, proof is offered that waves, their superpositions, and their rotations can all be easily expressed in terms of Euler's formula (which is itself proven in the text). A short derivation of the Schrödinger wave equation is also provided. Other topics in this chapter include: a proof that fundamental particles and wave packets are equivalent, a proof that the speed of light in a vacuum is constant, a proof of Einstein's famous equation that energy equals mass times the speed of light squared, and proof that photons are massless but that massless light carries energy and momentum. An understanding of physical waves leads to an understanding of how all known physical interactions unfold.

DOI: 10.1201/9781003516378-2

Related papers

Obtaining the Klein-Gordon wave equation without using the quantum operators

Roshan Raj

A derivation of the Klein-Gordon wave equation without substituting the corresponding quantum mechanical energy and momentum operators in Einstein's relativistic energy-momentum relationship is presented. An additional discussion is made to motivate readers to understand the distinction between the quantum wave equation and the governing law in the context of quantum mechanical formalisms. A resemblance of mathematical structure between the Maxwell wave equation and the deduced Klein-Gordan equation is shown. It is also noted that the wave velocity of the matter wave becomes equal to photon velocity for massless cases. This can be useful to explain why light equally shows matter and particle behaviour.

Oct 30, 20230 citations

The quantum wave equation

Daniel A. Fleisch, Laura Kinnaman

This is the third of three chapters that apply the concepts of Chapters 1, 2, and 3 to the three principal types of waves: mechanical, electromagnetic, and quantum. Although mechanical waves may be the most obvious in everyday life, and electromagnetic waves may be the most useful to our technological society, a case can be made that quantum waves are the most fundamental. As you'll see in this chapter, every bit of matter in the Universe behaves like a wave under certain circumstances, so it's hard to imagine anything more fundamental. If you're thinking that life in the macroscopic world moves along just fine without considering quantum effects, you should realize that the laws of classical physics are approximations, and all the quantum weirdness described in this chapter blends smoothly with classical physics when they overlap. Much of that weirdness comes from the dual wave and particle nature of matter and energy, and one of the goals of this chapter is to help you understand what's actually doing all this waving. This chapter begins with a comparison of the characteristics of waves and particles in Section 6.1, followed by a discussion of wave–particle duality in Section 6.2. You can read about the Schrödinger equation and probability wavefunctions in Sections 6.3 and 6.4, and quantum wave packets are discussed in Section 6.5. Wave and particle characteristics Before studying modern physics, most students think of particles and waves as belonging to fundamentally different categories of objects. That's understandable, because particles and waves have several very different characteristics, including the way in which they occupy space, the way they travel through openings, and the way they interact with other particles or waves. Here's a summary of some of the differentiating characteristics between particles and waves. Occupying space. Particles exist in a well-defined amount of space; if you could pause time at some instant during a tennis match, all of the spectators would agree on the position of the ball at that instant.

Cambridge University Press eBooksFeb 28, 20150 citations

Perceptions of the Schrodinger Equation

Spyros Efthimiades

The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.

arXiv:1307.6108Bulletin of the American Physical SocietyJul 23, 20130 citations

Introduction

Sönke Johnsen

This introductory chapter provides an overview of light. In particle language, a beam of light is a stream of photons—small massless particles with energy and momentum that travel at high speeds. In wave language, a beam of light is a series of waves of changing electric and magnetic field strength that have phase, amplitude, and wavelength, and—like photons—travel at high speeds. To intuitively describe what light does, one has to jump back and forth between these two interpretations. Sometimes, the results of experiments with light are more easily predicted or intuited using the mathematics and metaphors appropriate to waves; and sometimes the results are better explained using the mathematics and metaphors appropriate to particles.

Princeton University Press eBooksDec 25, 20110 citations

Waves and Particles

David Reginald Tilley

It has already been noted (section 2.8) that high frequency electromagnetic radiation can exhibit particle-like behaviour. For example, in the sensitive type of detector known as a photomultiplier tube (described in Wehr and Richards, Physics of the Atom , 2nd edn, Addison-Wesley, section 10.5), a beam of low-intensity X-rays is detected as a sequence of isolated sharp pulses, such as would be produced by a stream of particles arriving at the detector. As was pointed out, it is not surprising that it is usually high-frequency radiation which travels in particle form; as shown in figure 2.39, a wave group of high-frequency radiation can be compressed into a very short time span. However, when we speak of a particle, we mean something more than an entity which arrives in a short time span. A particle has a definite size, or more exactly it has a well-defined energy and momentum. It was discovered around about 1900 that the same is true of an X-ray wave packet: all packets of a given frequency radiation carry the same energy and momentum. This revolutionary discovery means that we must treat the wave packets as particles on exactly the same footing as more familiar particles such as electrons. The electromagnetic particle is called a photon . There is a wealth of experimental evidence which may be used to define the properties of a photon. We shall be highly selective, however, and simply present two particularly clear-cut experiments, the photoelectric effect and Compton scattering, which enable one to define the energy and momentum of the photon in terms of the frequency and wave number of the corresponding wave. These experiments will be described in detail in section 4.1.

Jan 1, 19740 citations

Derivation of the Schrodinger equation from fundamental principles

Wenzhuo Zhang, Anatoly Svidzinsky

Schrodinger path to the quantum mechanical wave equation was heuristic and guided more by physical intuition than formal deduction. Here we derive the Schrodinger equation for the particle wave function, assuming that it has a meaning of the probability amplitude to find the particle at time t at point r and the relations E=hw, p=hk expressing particle energy and momentum in terms of the frequency and wave vector of the associated probability wave.

quant-pharXiv:2603.27041Mar 31, 2026