Modern Physics Problem 10.3.1: Infinite Well

Description

This Physlet shows the solution to Schroedinger's equation for a particle inside an infinite square well.  It is solved using the "shooting method" in which an initial guess for the energy is made. After each iteration, compared to the known boundary value and the energy is refined an acceptable tolerance level is reached.  In this problem the trial solution to the wavefunction is calculated from left to right.

The boundary conditions for this problem, in general, are that:

  1. Y(-) = Y(+) = 0

  2. Y'(-) = Y'(+) = 0

  3. Y and Y' are continuous at the sides of the wells.

There are several important features to realize when presenting this exercise to students:

  1. Students can be confused by the plotting of the wave function with the horizontal axis shifting to the energy eigenvalue. To them this may mean that the wavefunction didn't go to zero at the boundary. Most books do this in order to compare the different wave functions.  See Section 18.3 for an alternate plotting method using data connections that does not shift the wavefunction.

  2. The walls of the graph are hard, i.e., the potential at the walls is always infinite.  This will have important implications in other exercises where new potentials are defined within the walls of the graph.

  3. It will always be possible to get a mathematical solution to the differential equation, but the important question for a physicist is "Does the solution have physical meaning?"  Solutions will have physical meaning if they satisfy the boundary conditions.

  4. 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.

Question

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?

Question

In order to keep from having to deal with very small and very large numbers, computer simulations often set a combination of constants in Schroedinger's equation to 1.  Using your answers to the first question and the theoretical values for the energy levels of an electron in the well, determine this scaling combination and the units of energy.

Question

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.

Question

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?

Credits

Physlet problems authored by Dan Boye.