This lab exercise will examine the different facets of a single particle in a square-well potential. Schroedinger's equation for this problem is solved using the "shooting method" in which initial guesses for the boundary conditions for the wave function, Y, and its first derivative, Y', are made. After an iteration, the guesses are refined and the results compared until an acceptable tolerance level is reached. In this problem the guesses are made for the left side of the graph and the solution is calculated from left to right.
The boundary conditions for this problem, in general, are that:
Y(-¥) = Y(+¥) = 0
Y'(-¥) = Y'(+¥) = 0
Y and Y' are continuous at the sides of the wells.
There are several important features to realize before starting this exercise.
The walls of the graph are hard, i.e., the potential at the walls is always infinite. This will have important implications once a new potential is defined within the walls of the graph.
It will always be possible to get a mathematical solution to the differential equation, but the important question for the physics community is "Does the solution have physical meaning?" Solutions will have physical meaning if they satisfy the boundary conditions.
Left-click in the graph for graph coordinates.
Right-click in the graph to take a snapshot of the current graph.
Left-click-drag the mouse inside the energy level spectrum to change energy
levels and wave function of the particle.
Exercises
What is the width of the above well? (The horizontal axis is in meters.) What are the energy levels for the first 6 energy levels? What functional dependence of the energy level on the quantum number do your results indicate?
In order to save the computer from having to deal with very small and very large numbers, some combination of the constants in Schroedinger's equation has been set to 1. Using your answers to the above exercise and the theoretical values for the energy levels of an electron in the well, determine this scaling combination and the units of energy.
Note where x = 0 is located in comparison to the infinite square-well solution in your text. Does this difference affect the energy levels and/or the wave functions? Explain.
The "parity" of a wave function is defined to be:
even if Y(x)
= Y(-x)
odd
if Y(x)
= - Y(-x)
What
is the parity of each of the wave functions for the first 6 energy levels?
What general conclusion can you draw regarding the quantum number and the
parity for an arbitrary energy level?
The Eigenvalue Physlet was written by Cabell Fisher, and Wolfgang Christian. This lab was prepared by Dan Boye.