This lab exercise will examine the different facets of a single particle in a square-well potential. Schrodinger'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.
Exercises
In order to save the computer from having to deal with very small and very large numbers, some combination of the constants in Schrodinger'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?