Shrodinger equation interactive solver

Developed by Dany Regelman

Introduction

The Shrodinger equation is a second order linear differential equation, which describes the wavefunction of a particle in a given potental V(z):

This equation has several solutions.

The program

Sdemo.m is a MATLAB function (requires MATLAB5 or higher) utilizing a graphical user interface capabilities. It solves the Shrodinger equation by finite differences method, which is the most efficient method for solving this type of boundary condition problems.

There are several predefined potentials : square finite barrier well , double finite barrier well and parabolic potential. In addition there is a possibility to draw your own potential by defining node points by clicking the mouse on the plot window.

The potential can be changed by moving horizontal lines :

Similarly the dimensions can be changed by moving the vertical lines:

This below sequence of pictures is an example of creating an arbitrary potential using the mouse. Particle mass (in the free electron mass units), length of the potential (in angstroms) , number of required states and the boundary conditions (fixed or periodic) can be changed by typing the values in the appropriate windows and pressing the “Enter” key. The new solution will appear on the screen automatically.

What to do:

1) For a single finite well count a number of confined states. Find out how the change of a particle mass affect the number of confined states. Do the same by changing the depth of the well.

2) For a double well observe the “bonding” and “anti-bonding” states. Check how the energy separation between them is affected by the barrier height and width.
3) Change the depth of one of the wells in a double well structure and observe how the solutions change. Change the well depth until the first state of well 1 become degenerate with the second state in well 2. Observe “bonding” and “anti-bonding” between different states.
4) Check the effect of boundary conditions on different potentials. Note, that the boundaries affect mostly a non confined states.
5) Draw your own potentials .