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One-dimensional time-independent Schroedinger equation; step-wise potentials

This site was put together by Eric Nodwell. You can find my email address on the MBE Lab web site contact list.

There are (at least) two possible approaches to numerically solving the one-dimensional time-independent Schroedinger equation for arbitrary potentials; you can either discretize the wavefunction or the potential. This solver discretizes the potential. This has the advantage of being exact for a large class of potentials that occur in practice, particularly for structures grown by MBE (e.g. quantum wells). Or at least it would be exact, if MBEs laid down a smooth continuous gunk instead of layers of discrete atoms. Using an appropriate effective mass is a crude order correction for discrete atoms.

Layers : d (nm), U (V), m (me)

Enter the layers in a spreadsheet format. The first column is for layer thicknesses, in nm, and the second potential, in volts and the third for effective mass, in units of free electron mass me. Columns can be separated by spaces or tabs. The outer layers are the semi-infinite media on either side, so you must enter at least three layers for any structure with discrete modes. The thickness has no meaning for the outer layers - just put zero.
Eigenvalues Wavefunction
E:V

The wavefunction calculation is only meaningful if you enter an energy which is an eigenvalue. Actually, you probably won't even need to bother with the wavefunction calculation from this page, because at the bottom of the eigenvalues results page, you will find links to generate the first few eigenfunctions.

Examples

Cut and paste these examples into the above box.
Single Quantum Well 
0 1 1
1 0 1
0 1 1
Step-wise approximation to harmonic potential 
0 1.25 1
0.2 1.058 1
0.2 0.882 1
0.2 0.722 1
0.2 0.578 1
0.2 0.45 1
0.2 0.338 1
0.2 0.242 1
0.2 0.162 1
0.2 0.098 1
0.2 0.05 1
0.2 0.018 1
0.2 0.002 1
0.2 0.002 1
0.2 0.018 1
0.2 0.05 1
0.2 0.098 1
0.2 0.162 1
0.2 0.242 1
0.2 0.338 1
0.2 0.45 1
0.2 0.578 1
0.2 0.722 1
0.2 0.882 1
0.2 1.058 1
0 1.25 1
Two coupled QWs 
0 1 1
1 0 1
0.5 1 1
1 0 1
0 1 1
Three coupled QWs 
0 1 1
1 0 1
0.5 1 1
1 0 1
0.5 1 1
1 0 1
0 1 1

Source Code

The source code itself is qwells.py; you will also need utils_qwells.py.

If you find this site useful, I encourage you to download the source and run the calculations locally. This will have several advantages. Among them are that you will have much more control over the calculations and outputs.

Legal

You use this website entirely at your own risk. Improper use of this website may contravene the Second Law of Thermodynamics, and consequently result in the automatic rejection of your patent application. If you want your devices to operate as designed, you should independently verify any results provided here.

If you download any source code files, please read and adhere to the license at the top of each source file. It's not too onerous, really.

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