This lab exercise will examine the different facets of a single particle in an infinite square-well potential. Refer to Section 6-5 of your text for a theoretical development of the problem.
There are several important features to realize before starting this exercise.
1. The walls of the graph are hard, i.e., the potential at the walls is always infinite.
2. 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:
Y(-¥) = Y(+¥) = 0
Y'(-¥) = Y'(+¥) = 0
Y and Y' are continuous at the sides of the wells.
3. 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:
Section 1 (Initialize)
What are the energies 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. What is the width of the well? (The horizontal axis is in meters.) Using this width and the theoretical values for the energy levels of an electron in the well, determine this scaling combination and the units of energy.
Measure the wavefunctions by left-clicking in the graph. Do your measurements agree with the solutions in your text? Are they normalized?
Display Plot Y^{2}. Y^{2}(x)dx is the probability of finding the electron in the range x to x+dx. Remember that the probability of finding the electron in the box is 1. What is the probability of finding the electron in the left half of the box for the first 6 energy levels? For n=4, what is the probability of finding the electron in the range from x=0.25 to 0.5? From x=0.25 to 0.825? From x=0.1 to 0.35?
Section 2 (Initialize)
The Eigenvalue Physlet was written by Cabell Fisher, and Wolfgang Christian. This lab was prepared by Dan Boye.