Optional extra tutorial

Some students have asked for extra feedback and advice on passing the course.
Tomorrow at 1pm I will be available in the interaction room. If you wish to work through tutorial problems and/or past exam questions, and get feedback I will be available then. If it is popular enough I will do this every week, at a mutually agreeable time.

Exam practice and feedback

Several students have asked for concrete suggestions on improving their performance on the final exam. I suggest the following exercise.

Write out a new formula sheet.
Take the mid-semester exam and try and do it again in one hour by yourself.
Give me your written solutions and I will mark it as I would normally.
Note this will not change your mark, but hopefully it will help you get a better grasp of the material, and get more feedback of what I expect.

Hartree Approximation

The Hartree approximation arises from a first attempt to account for the effects of electron-electron interactions, which are neglected when using the independent electron approximation (as in the bloch model etc). The single-electron wavefunction seems insufficient to properly describe how the electrons in an N-particle system interact. So the Schrodinger equation for an N-particle system is considered, i.e. we want a wavefunction for the N electron system. However, attempts to solve this equation prove to be futile.

Instead, the single electron wavefunction that best represents the interactions is analysed, specifically, a potential that includes the potential of the ions U-ion and the electric fields from the other electrons in the system U-el (the approximation is made that this is given by a smooth distribution of negative charge). The contributions from all the electrons to the potential energy of the electron considered can then be worked out (A&M pg. 330-331) and plugged into the Schrodinger equation for that particular electron. If such an equation is written for each electron, the set of equations is known as the Hartree Equations. The Hartree equations are solved computationally by proposing a form of the potential U-el, solving the equation to obtain the wavefunction, from which the next U-el is found, and so forth until the potential reaches a certain degree of accuracy (does not change very much between successive iterations).

Energy bands

Here is a useful Pdf for better understanding of energy bands and periodic potential.

http://www.teknik.uu.se/ftf/education/ftf1/forelasningar/overview/energybands.pdf

Lecture slides on electron-electron interactions

Here is are the draft slides for the next few lectures.
At the tutorial on wednesday I will work through the answers to the mid-semester exam.

NFE model

Here is a very useful pdf for a nearly free electron model. (NFE)

http://www.cmmp.ucl.ac.uk/~ahh/teaching/3C25/Lecture19s.pdf

Summary of tight binding model

As it appears from the model name, in this approximation electrons are tightly bind to their atoms with limited interaction with other atoms in their neighbor. this approximation minimise the overlap. but small correction in Hamiltonian is needed.