Quantum Mechanics and Solid State Physics II

Abstract

Solution and application of the Schrödinger equation for an electron in a Coulomb potential (the H atom problem). Energy bands in solids from the Nearly-Free-Electron and Linear-Combination-of-Atomic-Orbitals approaches. Band gaps, semiconductors and insulators. Solution of the Schrödinger equation for a harmonic oscilattor; vibrational properties of solids

Objective

By the end of this semester you will be able to: Understand the solution of the Schrödinger equation describing a H atom and apply it to calculating properties of atoms. Derive how energy bands in solids emerge both from combining atoms and from applying a periodic potential to free electrons. Construct band structures for given crystal chemistries, and interpret given band structures to predict material properties. Solve the Schrödinger equation for the harmonic oscillator and apply it to the vibrational properties of solids. Explain the relationships between atomic orbitals, chemical bonds, crystal structures and resulting properties of matter.

Content

By the end of last semester we had developed the classical and quantum mechanical free-electron theories, and found that, particularly by including quantum mechanics, we could describe many of the properties of simple metals quite well. We found some peculiarities though, like positive Hall coefficients and negative masses, and didn't have the machinery to describe semiconductors or insulators. We'll start this semester by developing the band theory of solids, in order to address some of these deficiencies. We'll take two approaches: Starting from our Quantum Mechanical Free Electron Theory we will add a periodic potential, so that the electrons are not "free" any more. And starting from solving the Schrödinger equation for an atom, we will bring atoms together to form a crystal so that their atomic orbitals interact. We'll see that both approaches give us bands of allowed energy levels, with gaps between them, and allow us to capture many of the electronic properties of semiconductors and insulators. Finally, to understand thermal properties we will need to look at lattice vibrations, which will involve solving the Schrödinger equation for a harmonic oscillator.

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