Exercise 2.1-1

Quick Questions to:

2.1 Basic Band Theory

Here are a few quick questions to 2.1.1: Essentials of the Free Electron Gas
What happens, if you do not choose U = U0 = 0 but U = U1 ?
What does the sentence "...a plane wave with amplitude (1/L)3/2 moving in the direction of the wave vector k" mean"? Wave vectors, after all, are defined in reciprocal space with a dimension 1/cm. What, exactly, is their direction in real space?
Recount what you know bout the spin of an electron.
Where does the (1/L)3/2 in the solution of the Schrödinger equation come from? What would one expect for a crystal with the dimension Lx, Ly, Lz?
What kind of information is contained in the wave vector k?
Consider a system with some given energy levels (including possibly energy continua). Distribute a number N of classical particles, of Fermions and of Bosons, respectively, on these levels. Describe the basic principles employed..
How does one always derive the density of states D(E)?
Here are a few quick questions to 2.1.2: Diffraction of Electron Waves
Consider a fcc and bcc lattice with lattice constant a = 0.3 nm. Give the distance between {100} planes and the distance between the corresponding atomic planes. Do the same thing for the {111} plane of a fcc lattice with just one atom in the base, and for a diamond structure.
Remember the Ewald construction? Describe and explain for what kind of situations it is particularly useful.
Compare the free electron gas model with and without diffraction.
Here are a few quick questions to 2.1.3: Energy Gaps and General Band Structure
Draw a one-dimensional realistic periodic potential Now draw in the first Fourier component. Add the probability densities for finding electrons with k = kBZ. Explain the energy splitting and why DE is approximately given by the first Fourier component of the potential.

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