Quantum Mechanics 1
Schršdinger's Equation in One Dimension

Project: Electron states in a one dimensional lattice.

Electrons in a crystalline lattice possess special properties which makes solid state physics a distinctive branch of physics. These properties include the bunching of energy levels into bands, and this feature is crucial in the explanation of the differences between conductors, semiconductors and insulators.

Here we are only concerned with  quantum mechanical aspects of the problem. Although we can model the potential experienced by an electron in a regular crystalline solid as a set of neighbouring potential wells laid out in three dimensions, many of the features of interest can be investigated qualitatively by examining the properties of electrons in a set of one-dimensional wells. 

The fourth section of the CUPS software you have been using, entitled Electron States in a One-Dimensional Lattice, allows systematic investigation of features of interest. The number of wells may be varied in the range 1 to 12, and other variables include the shape of each well and the other parameters which specify it, e.g. width and depth. The software is easy to use; in particular many of the features are the same as those which you will have used earlier in investigating bound states of single potential wells.

The aims of the project, which is to be undertaken by students in teams of three (occasionally four), are as follows:  (a) to explore the software and to gain a quick appreciation of the results which it can generate; (b) to identify features of interest which  you would like to investigate in more detail; (c) to carry  out that work programme, and (d) to  write up the results. If you need any help as regards initial inspiration, you may like to look over the photocopy of the relevant chapter of the book which accompanies the software, which is available on short term loan in the library. Beware however of simply taking the list of exercises provided and systematically working through them. This is extremely unlikely to produce a satisfactory result in the time available. 

A list of group members in each team is on the noticeboard. Teams should aim to spend about 6 hours each on the project, including time to write up the results. The report should be word-processed, for ease of assessment, and be between 1500 and 2500 words in length. Any reports which are substantially longer than 2500 words (excluding diagrams etc.) will be penalised. You should avoid spending an undue amount of time on fancy presentational aspects. Marks will be awarded mainly for the physical content of the report, and slickness in presentation (or the opposite) will make only a marginal difference.

The reports will be marked by Dr Gillies or myself. It is intended that the assessment mark for the group should be modulated to take into account the actual contribution of group members, through information to be provided by group members themselves.

Reports should be submitted to me by 12 noon on Friday 22 November. The authors of late submissions will be penalised.

N C McGill
31 October 1996