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arXiv:1609.07359 · Canadian Journal of Physics

Explaining electric conductivity using the particle-in-a-box model: quantum superposition is the key

Umaseh Sivanesan, Kin Tsang, Artur F. Izmaylov

Published Jun 1, 2017• 0 citations
arXivHTMLPDF

Abstract

Most of the textbooks explaining electric conductivity in the context of quantum mechanics provide either incomplete or semi-classical explanations that are not connected with the elementary concepts of quantum mechanics. We explain electric conductivity using the simplest model system in quantum mechanics, a particle in a box (PIB). To stimulate the particle dynamics, a linear potential tilting the bottom of the box is introduced, which is equivalent to imposing a constant electric field for a charged particle. Although the PIB model represents a closed system that cannot have a flow of electrons through it, we consider the oscillatory dynamics of the particle probability density as the analogue of the electric current. Relating the amplitude and other parameters of the particle oscillatory dynamics with the gap between the ground and excited states of the PIB model allows us to demonstrate one of the most basic dependencies of electric conductivity on the valence–conduction band gap of the material.

DOI
10.1139/cjp-2017-0158

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Most of the textbooks explaining electric conductivity in the context of quantum mechanics provide either incomplete or semi-classical explanations that are not connected with the elementary concepts of quantum mechanics. We explain electric conductivity using the simplest model system in quantum mechanics, a particle in a box (PIB). To stimulate the particle dynamics, a linear potential tilting the bottom of the box is introduced, which is equivalent to imposing a constant electric field for a charged particle. Although the PIB model represents a closed system that cannot have a flow of electrons through it, we consider the oscillatory dynamics of the particle probability density as the analogue of the electric current. Relating the amplitude and other parameters of the particle oscillatory dynamics with the gap between the ground and excited states of the PIB model allows us to demonstrate one of the most basic dependencies of electric conductivity on the valence–conduction band gap of the material.
Chapter 3 provides a brief review of a few concepts from classical Newtonian physics, and how this fails to explain phenomena at the nanoscopic scale. As is detailed within, the classical mechanical model is able to describe all observed physical phenomena; however, when it comes to the nanoscale this model breaks down, because the properties of the materials change dramatically. At this scale, the surface-to-volume ratio increases, gravitational forces become negligible and random molecular motion becomes more important. This changes the properties of the material and affects the devices that are fabricated at this scale. It is hence necessary to develop a new model that can explain such physical phenomena. This has led to a new branch of physics, i.e. quantum physics (QP) or quantum mechanics, which is able to explain the behavior observed at nanoscale. QP enables us to explain phenomena such as the photoelectric effect, Compton effect. etc. In this chapter, we introduce quantum mechanics and how it evolved in the late 19th and early 20th centuries. Further, light’s behaviour as a particle as well as a wave is described by the photoelectric effect and Young’s double-slit experiment, respectively. Further, the dual nature of the particle is explained using the de Broglie principle. Schrodinger’s equation is then explained, which gives the probability of finding a subatomic-scale particle at a certain position in a quantum well. In addition to the formulation of Schrodinger’s equation, its solution, the concept of a potential well, the characteristics of the wave function and its interpretation and significance, and, finally, the application of Schrodinger’s equation in the structure of an atom form subsequent parts of this chapter. The equation serves as a basis with which to understand band theory. Band theory explains the difference between conductors, insulators and semiconductors. Materials are distinguished according to the differences in their band gaps. Adding impurities or varying temperature changes the band gap in semiconductors, which consequently changes the properties of the semiconductor and its electrical conduction. One more important measure to understand and exploit the properties and behaviour of materials is Fermi–Dirac statistics. It is useful to acquire the probability of electrons at a given energy level at a specific temperature. These concepts can be utilised by physicists and chip makers to make more energy-efficient devices. Hence, this chapter details the fundamentals of quantum physics and how they relate to the electrical properties of nanodevices.
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Today it still remains a challenge whether quantum mechanics has an underlying statistical explanation or not. While there are and were a lot of models trying to explain quantum phenomena with statistical methods these all failed on certain levels. In this paper a method is proposed that is not only based on a classical statistical framework but it has an underlying physical model behind it and it can explain some of the basic characteristics of quantum mechanics. It will be shown that if look at the properties of a charged particle in a random electric field one can obtain the discrete energy values of the harmonic oscillator and the infinite potential well, and also gives a good qualitative description of the double-slit experiment and measurement theory. Also the side-effect of the model is the emergence of a constant with an action dimension.