# Homework #5, 2013

*Due on last day of class (Friday) at 5pm*

## Q1: Use wavenumber to sketch LCAO states

Problem 15.5 from McIntyre.

*Note: The problem asks “what is the energy of this state”. You can express the energy in terms of α and β.*

## Q2: Calculation of the overlap integral, β

Problem 15.8 from McIntyre.

*Note: Section 15.8 of the text book tells you the ground state energy of this particular isolated potential well. You can also use the band structure software to check the energy. This will save some time when determining the ground state wavefunction (see Chapter 5).*

## Q3: Use density of states to calculate total energy

Problem 15.6 from McIntyre. Just the second part of the problem: “show that the integral evaluates to *N*α despite the infinity in *g*(*E*)”.

## Q4: Density of states for a free particle

Problem 15.7, part a, from McIntyre.

*Note: This is a tricky problem to conceptualize. How do I think about a “free particle” that is subject to boundary conditions? Imagine an electron traveling around a circular particle accelerator. The electron comes back to where it started after traveling a distance L. If the electron is in an energy eigenstate, the probability of finding the electron at a certain location is equally distributed around the circle. The eigenstate wavefunction must be single-valued (only one value for each position), therefore, the wave vector that is part of the eigenstate wavefunction is restricted to certain values.*