We discuss the breakdown of classical theory in relation to phenomena on the nanoscale. The historical discovery of the wave nature of electrons in the Davisson–Germer Experiment is reviewed. We present the puzzling experimental data and its explanation in terms of particle diffraction, which contradicts classical mechanics. The quantitative success of de Broglie’s formula in associating particle momenta with a wavelength is demonstrated. Analyzing the conditions in which the wave nature of particles becomes apparent, namely, the condition for correspondence between the de Broglie wavelength and the lattice from which the particles are diffracted, we draw some general conclusions. Particularly, by translating to de Broglie wavelengths the particle masses and energy values that are typical to materials and processes on the nanoscales, one immediately realizes that wave properties are expected to be dominant. Quantum mechanics is therefore essential for a proper description of nanoscale phenomena.
We discuss the breakdown of classical theory in relation to phenomena on the nanoscale. The historical discovery of the wave nature of electrons in the Davisson–Germer Experiment is reviewed. We present the puzzling experimental data and its explanation in terms of particle diffraction, which contradicts classical mechanics. The quantitative success of de Broglie’s formula in associating particle momenta with a wavelength is demonstrated. Analyzing the conditions in which the wave nature of particles becomes apparent, namely, the condition for correspondence between the de Broglie wavelength and the lattice from which the particles are diffracted, we draw some general conclusions. Particularly, by translating to de Broglie wavelengths the particle masses and energy values that are typical to materials and processes on the nanoscales, one immediately realizes that wave properties are expected to be dominant. Quantum mechanics is therefore essential for a proper description of nanoscale phenomena.
This chapter discusses the conceptual foundations of quantum mechanics in the context of matter. Classical physics identified matter with particles and radiation with "waves." Although this description accounted for a large number of very important facts, it became apparent around the turn of the century that other phenomena, such as blackbody radiation and photoelectric effect, are satisfactorily explained only if radiation is described in terms of particles. Sometime later the hypothesis that, conversely, matter has wave properties was proposed by Louis de Broglie and was confirmed by the Davisson-Germer experiment of electron diffraction. In this experiment and similar ones, an electron beam was passed through a regular lattice (crystal), and subsequently showed the well-known diffraction patterns that are characteristic of waves. These results confirmed the theoretical predictions that a monoenergetic electron beam should be described by a plane wave and that the wave vector k of the wave should be related to the momentum p of an individual electron.
The fact that light has some of the properties of ordinary particles led Louis de Broglie, in 1924, to construct a theory on the proposition that photons are particles. In the theory of wave motions, there is in general a difference between the speed of waves of some uniquely defined frequency, and the speed of a localized pulse such as de Broglie proposed to associate with a moving particle. Most often, one wishes to know the de Broglie wavelength for particles of a given kinetic energy. C. J. Davisson and L. H. Germer deliberately used single crystals as targets, and proceeded to discover just the kind of behavior that de Broglie's hypothesis of particle waves predicted. The development of electron microscopy, with its use of extremely well collimated, highly monoenergetic electron beams, has led to some beautiful demonstrations of the wave properties of electrons in analogs of optical diffraction phenomena. The strong reflection is a wave-like phenomenon totally unexpected from the classical picture of electrons.
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.
The chapter presents the first application of wave ideas to quantum particle descriptions, including describing the Davisson–Germer experiment, in which electrons were diffracted as waves from crystals. The work of famous early developers of quantum mechanics—Planck, Einstein, and de Broglie—led to relating momentum to wavelength, and energy to frequency. The wave function and its probability interpretation by Born are introduced. The wave functions of a particle in a box potential are exactly analogous to classical standing waves, giving these matter waves quantized energies just as frequencies are quantized in the harmonic series of a stringed instrument. The Heisenberg uncertainty principle follows from the general wave analysis. Zero-point motion and zero-point energy are important concepts resulting from the application of wave ideas to particles.
Many breakthrough experimental discoveries in the realm of physics are described, notable among them being Young’s double-slit experiment, Davisson and Germer experiment on electron diffraction, and Crompton scattering of X-rays or γ -rays. Also explained are revolutionary theories such as Planck’s quantum hypothesis, Bohr’s model of atomic structure, and Einstein’s interpretation of photoelectric effect, which laid down the foundations of quantum theory. Wave-particle dualism is epitomized in cornerstone theories namely de Broglie’s hypothesis and Heisenberg’s uncertainty principle. Surprisingly, the old notion of treating electrons as particles moving in an electric field works satisfactorily in many everyday-life applications but leads to serious errors at the nanoscale where the wave picture of electrons salvages our understanding of matter and energy. Electrons can act as particles or waves. The wave picture has its origins in the quantum theory.
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.