HW4_FC

Determining the Energy Levels of an Electron in a Square Quantum Well (SQW)

Objective

Using the program developed in the lab session, determine all the energy levels of the electron in a Square Quantum Well (SQW) for well widths ranging from 5 nm to 100 nm, in increments of 5 nm. Refine the energy discretization where necessary. Construct a graph that shows the variation of the energy levels with the well width (well width on the x-axis, and energy on the y-axis). The 1st, 2nd, 3rd, 4th, etc. levels will be represented by different-colored, monotonically decreasing curves. The higher the energy level, the higher the corresponding curve will start at larger well widths. Present the resulting figure on an A4 sheet and comment on the obtained results.

Implementation

1. Start with the program developed in the lab session, which can solve the Schrödinger equation for a Square Quantum Well (SQW).
2. Modify the program to accept well widths in the range of 5 nm to 100 nm, with increments of 5 nm.
3. Refine the discretization of the energy axis, where necessary, to ensure accurate representation of the energy levels.
4. For each well width, calculate the energy levels of the electron in the SQW.
5. Construct a graph with the well width on the x-axis and the energy levels on the y-axis.
6. Represent the 1st, 2nd, 3rd, 4th, etc. energy levels using different-colored, monotonically decreasing curves.
7. Ensure that the higher energy level curves start at larger well widths, reflecting the fact that higher energy levels require larger well widths to appear.
8. Present the resulting graph on an A4 sheet and provide a commentary on the obtained results.

Commentary

The resulting graph should show the following:

1. As the well width increases, the energy levels of the electron decrease, reflecting the inverse relationship between the energy levels and the well width.
2. The curves for the higher energy levels start at larger well widths, indicating that higher energy levels require larger well widths to appear.
3. The curves are monotonically decreasing, with the higher energy level curves starting at higher energies and converging towards the lower energy levels as the well width increases.
4. The spacing between the energy levels decreases as the well width increases, reflecting the fact that the energy level spacing becomes smaller for larger well widths.
5. The graph provides a visual representation of the relationship between the energy levels and the well width in a Square Quantum Well, which is essential for understanding the behavior of electrons in these confined structures.

By analyzing this graph, you can gain insights into the quantization of energy levels in a Square Quantum Well and how the well width affects the energy spectrum of the electron.