Quantum Mechanics and Solid State Physics I

Abstract

Classical theory of metals; successes and failures. Introduction to quantum mechanics. Solution of the Schrödinger equation for free electrons and application to describe the properties of metals. Basic quantum mechanical formalism and solution of the Schrödinger equation for other model potentials.

Objective

By the end of this semester you will be able to: Appreciate the remarkable successes and spectacular failures of the classical theory of metals. Motivate the need for a theory beyond classical mechanics to describe the properties of materials. Formulate and solve the Schrödinger equation for simple problems. Describe the properties of simple metals using quantum mechanical free-electron theory. Apply the formalism of quantum mechanics to calculate things that can be measured and to interpret physical processes.

Content

How well can we explain the properties of materials by treating the electrons as classical particles? Surprisingly well for simple metals, in fact, but not well enough to guarantee that our electronic devices function and our bridges do not fall down. To be rigorous and predictive, we need to use quantum mechanics, which we will introduce and learn how to use this semester. We'll see that, in its simplest form, the Schrödinger equation does a good job for straightforward systems like simple metals. We'll also find some failures that will motivate further developments next semester. We’ll start with the classical theory of metals and introduce the so-called Drude model of electrical conductivity in which the electrons are treated as classical particles. We’ll use this model to predict properties that we can measure, such as the Hall coefficient and the electronic heat capacity. We’ll see that sometimes the classical theory does OK, but in many cases it gets things completely wrong. It’s clear that we need a better theory. The better theory that is most widely used to describe the properties of materials is Quantum Mechanics. We’ll introduce the Schrödinger equation and look at what the quantum mechanical wave function means and how to work with it. Then we will solve the Schrödinger equation for “free electrons”, in which the quantum mechanical electrons don’t interact with anything. We’ll introduce concepts of band structure, density of states and the Fermi-Dirac distribution, and show that the quantum mechanical free-electron theory does better than classical Drude theory in describing the properties of metals. For anything more than simple metals, though, we’ll find that quantum mechanical free-electron theory is inadequate and we have to extend our description so that the electrons interact with the world through a potential. We’ll look at some model potentials corresponding to some simple situations that are relevant in materials science and also introduce some formalism and definitions that will make our lives easier later.

Resources