This chapter presents the double-slit experiment and Davisson-Germer’s corresponding experiment that proved the wave-like nature of the electron in 1927. Next, it shows the basics of the special theory of relativity which is used by de Broglie to obtain the wave-particle duality. Afterwards, it displays the most important properties of a periodic wave along with the concepts of travelling wave, composite wave, group velocity, wave packet and single wave packet. These sections are the basis for understanding the de Broglie’s work where he used the special theory of relativity, Einstein’s quantum theory of light and the features of the periodic waves to obtain his famous constant relation of the product of linear momentum (of a particle) and wavelength (of a wave) which became the most important quantum concept used by Schrödinger to develop the quantum wave theory. In the end, a do-it-yourself activity is also given in order to plot several wave packets. (153 words).
This chapter presents the double-slit experiment and Davisson-Germer’s corresponding experiment that proved the wave-like nature of the electron in 1927. Next, it shows the basics of the special theory of relativity which is used by de Broglie to obtain the wave-particle duality. Afterwards, it displays the most important properties of a periodic wave along with the concepts of travelling wave, composite wave, group velocity, wave packet and single wave packet. These sections are the basis for understanding the de Broglie’s work where he used the special theory of relativity, Einstein’s quantum theory of light and the features of the periodic waves to obtain his famous constant relation of the product of linear momentum (of a particle) and wavelength (of a wave) which became the most important quantum concept used by Schrödinger to develop the quantum wave theory. In the end, a do-it-yourself activity is also given in order to plot several wave packets. (153 words).
This chapter describes the dual nature of electron/de Broglie waves. de Broglie suggested that like light, an electron also behaves as both a particle and a wave. Based on this hypothesis, a relationship between wavelength of electrons (a particle) and momentum was developed. It opines that ‘wavelength of the particle is inversely proportional to momentum’. This is called the de Broglie relation. On the basis of the above facts, we have developed a relationship between wavelength and voltage applied and this relation may be stated as ‘the wavelength of a particle is inversely proportional to the square root of the voltage applied to the system’. This chapter also uses the Davisson and Germer experiment to show the wave nature of the particle, i.e., diffraction of electrons by crystal lattice. The experimental results were the same as found in the case of the diffraction of X-ray by crystals. This chapter also provides an idea on quantisation of angular momentum. It also discusses Heisenberg’s uncertainty principle, phase velocity, group velocity, and uncertainty relation between energy and time. The chapter comprises the experimental evidence of Heisenberg’s uncertainty principle, such as by diffraction of electrons through a slit and gamma ray microscope thought experiment, which is shows the physical significance of the uncertainty principle. Sufficient references, solved problems, and questions on concepts have been provided at the end of chapter.
Abstract This chapter examines some crucial experiments and basic principles of quantum mechanics that have strongly influenced the development of quantum theory in its modern form: starting from Planck’s law of black-body radiation and its derivation using the quantization of energy; Einstein’s photon hypothesis for the puzzle presented by the photoelectric effect and its test via the Millikan experiment; the Compton effect; Bohr’s theses for the structure of the hydrogen atom and the complementarity of waves and particles i.e., the wave-particle duality, culminating in de Broglie’s ideas for matter waves and their confirmation in the Davisson-Germer experiment for electron diffraction from a crystal; Heisenberg’s uncertainty principle for position and momentum as well as for energy an time; the famous double-slit experiment and Born’s probability interpretation, and finally the provocative thought experiment of Schrödinger’s cat
In 1924, de Broglie postulated that particles can behave like waves, thus complementing the observation by Einstein in 1905 that light can behave like particles. This wave–particle duality aspect for both particles and waves had a deep impact on the subsequent development of quantum mechanics. Some highly counterintuitive results, like the Heisenberg uncertainty relation and the Bose–Einstein condensation, that were motivated by wave–particle duality are discussed in this chapter. Following de Broglie’s hypothesis, a wave packet description for a particle is described. An analysis of the Heisenberg microscope is presented, thus motivating the Heisenberg uncertainty relation. The Davisson–Germer experiment that showed that electrons can behave like waves and the Compton effect that provided early conclusive evidence that light can behave like particles are also discussed.
Wave-particle duality, the revolutionary concept that particles can exhibit both wave-like and particle-like properties, was first proposed by Louis de Broglie. His equations connecting a particle's energy and momentum to the properties of its corresponding wave laid the groundwork for quantum mechanics. This fundamental principle explains the surprising results of the double-slit experiment and underlies many of quantum mechanics' unique behaviors. In this work, relying solely on Einstein's two postulates of special relativity, and the idea that time is single valued, we will demonstrate that a complete theory of relativity predicts a wave-like quantizable behavior for particle trajectories. We will prove that the theory of relativity predicts a first-order, spin-independent transport equation as an equation of motion for particles, in addition to the classical relativistic Hamiltonian. As key results, we will demonstrate that our theory provides an integrable solution for a free scalar particle, offering insights into the wave-particle duality observed in nature. For spinor particles, we will show that our theory generalizes Weyl theory to accommodate massive particles, while also allowing for modified Dirac-like particles.
Wave-particle duality, the revolutionary concept that particles can exhibit both wave-like and particle-like properties, was first proposed by Louis de Broglie. His equations connecting a particle's energy and momentum to the properties of its corresponding wave laid the groundwork for quantum mechanics. This fundamental principle explains the surprising results of the double-slit experiment and underlies many of quantum mechanics' unique behaviors. In this work, relying solely on Einstein's two postulates of special relativity, and the idea that time is single valued, we will demonstrate that a complete theory of relativity predicts a wave-like quantizable behavior for particle trajectories. We will prove that the theory of relativity predicts a first-order, spin-independent transport equation as an equation of motion for particles, in addition to the classical relativistic Hamiltonian. As key results, we will demonstrate that our theory provides an integrable solution for a free scalar particle, offering insights into the wave-particle duality observed in nature. For spinor particles, we will show that our theory generalizes Weyl theory to accommodate massive particles, while also allowing for modified Dirac-like particles.
Wave-particle duality is widely regarded as one of the conceptual pillars of modern quantum theory, providing the framework through which the behavior of microscopic matter can be consistently interpreted. According to this principle, microscopic entities such as electrons, photons, atoms and even complex molecules exhibit both wave-like and particle-like properties depending on the experimental configuration used to observe them. This duality is quantitatively formalized through the De Broglie relation, which associates a wavelength with any particle possessing momentum. The present work provides a comprehensive theoretical analysis of the physical and mathematical foundations of wave-particle duality, focusing particularly on the De Broglie wavelength and the kinetic energy of the electron in both non-relativistic and relativistic regimes. The connection between the De Broglie hypothesis and Schrödinger’s equation is discussed in detail, highlighting the role of the kinetic energy operator and the interpretation of matter waves in terms of quantum wave packets. Experimental evidence supporting the wave nature of matter, including electron diffraction experiments and interferometry techniques, is also reviewed. Beyond their conceptual importance, these principles underpin several modern technologies, including electron microscopy, nanoscale semiconductor devices and quantum metrology. The analysis demonstrates that the De Broglie relation provides a unifying framework connecting momentum, wavelength and kinetic energy, thereby forming a fundamental bridge between classical mechanics, quantum theory and modern technological applications.