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:
Y(-¥) = Y(+¥) = 0
Y'(-¥) = Y'(+¥) = 0
Y and Y' are continuous at the sides of the wells.
Note the following points :
The wave function is plotted with it's "zero line" rising along the vertical axis (to correspond to the shifting to the energy eigenvalue).
The walls of the graph are hard, i.e., the potential at the walls is always infinite.
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.
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.
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 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.
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?
How do the spacing of the energy levels compare here to the finite square well energy levels, or to the quantum harmonic oscillator?
Related physlets : infinite square well and the finite square well
Credits : Physlet problems authored by Dan Boye. Modified by Scott Schneider.