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Free Electrons in QuantumMechanics

J. Hoekstra

Published Oct 14, 2016• 0 citations
Publisher

Abstract

In quantummechanics electrons are described both as particles and as (matter) waves. Typical quantum mechanical phenomena such as energy quantization and tunneling are possible due to the wave nature of the electron. In the next two chapters an introduction to quantum mechanics is presented tailored for our needs. Especially, in this chapter the focus is on the quantum mechanical description of free electrons.

DOI
10.1201/9781315364483-10

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In quantummechanics electrons are described both as particles and as (matter) waves. Typical quantum mechanical phenomena such as energy quantization and tunneling are possible due to the wave nature of the electron. In the next two chapters an introduction to quantum mechanics is presented tailored for our needs. Especially, in this chapter the focus is on the quantum mechanical description of free electrons.
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3.4. SummaryIn this Chapter, we have shown the limitations of classical mechanics and the success of quantum mechanics. The basic concepts and formalism of quantum mechanics have been exposed, including the quantized nature of the electromagnetic field, the wave-particle duality, the probability of presence of a particle, the wavefunction, and the Schrödinger equation. Simple quantum mechanical systems have been analyzed to understand these novel concepts, including an infinite and a finite potential well. Through these, some aspects of quantum mechanics have been discussed, including the quantization of energy levels and momenta, and tunneling effects.
This chapter is an introduction to the study of the quantized nature of matter. It begins with an introduction to the line spectrum produced by energized gases and the Bohr model that was created to explain the hydrogen spectrum. Emphasis is put on the energy analysis of the Bohr model to connect back to the theme of this volume of the text. The wave nature of matter is introduced as a way to explain the assumptions that Bohr made in his model and how these concepts led to the development of quantum mechanics. This chapter concludes with an introduction to quantum mechanics as an expression of the energy of a matter wave and as a way to explain atomic orbitals.
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.
Abstract—One of the most interesting and peculiar phenomena of Quantum Mechanics is the interference (I) induced by the electrons. Strangely enough, though the electrons are real particles, they often behave just like waves. From the point of view of the classical mechanics the I induced by the electrons is unexplainable, however it is solved mathematically using the formalism of quantum mechanics and applying Schrödinger’s equation. The quantum solution of the problem is clear and elegant, especially from a mathematical point of view, however it still leaves some perplexities as to understand how exactly the phenomenon happens. We will make a hypothesis trying to understand the undulation phenomenon of the electron: it is really a strange and mysterious phenomenon. Maybe if we consider that the electron, just as the baryons and the mesons, might be made of smaller particles (saving the integrity of the unity of the negative electrical charge and the other Laws of Conservation), we could understand more easily how a single electron can go through two close holes at the same time. Analogously we could better understand another very particular quantum phenomenon carried out mainly by electrons, that is the tunnel effect. In this case, though the particle does not have enough energy to go through the potential barrier, though it does not have any material possibility to pass through a layer which does not have any hole, after several “attempts ” the particle will manage to pass through the barrier anyway, as it had dug a tunnel, or as it had managed to find a “breach ” in the wall. In this phenomenon too, though we can explain it from a mathematical point of view, using the equations of the quantum mechanics, it is still not clear how actually the electron manages to have an undulation behaviour.