Modern Physics Problem 10.3.2: Finite Well

Description

In exercise is designed to study the finite square well and to show the shooting method works. Although the finite square-well potential problem is more realistic then the infinite well, it is difficult to solve because it yields transcendental equations.  With the finite potential, it is possible for the particle to be bound or unbound.   A bound level is one whose energy is less than the well depth.

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 vertical 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 the Chapter 18 section 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

Explain how you can observe that an energy value is acceptable as you right click-drag.

Answer: Right-click and drag in the energy spectrum and you should observe the right side of the wave function flipping from negative to positive.  This is a sign that you have passed through a physical solution to Schroedinger's equation, i.e., that the boundary condition at Y( x = +/- 1.00) =0. The correct boundary condition is assumed at the left side of the graph and the wavefunction is calculated in the direction of increasing x.

Question

By changing the principal quantum number, determine the bound state energies for this well.  How do the energies corresponding to the same quantum number compare for finite and infinite potential wells?

Question

Determine the number of  bound states for this well.  A solution of Schroedinger's equation for this problem indicates that the total number of bound states is the next largest integer above the product of the width divided by pi and the square root of the depth.  (Note the scale of the vertical axis in the graph.)  Does this hold true for your results?  Identify in the equation for the potential the parameter that determines the width of the well and make it half as wide.  Does the number of bound states still equal the predicted value?

Question

Decrease the depth of the well to 200 units while keeping the bottom of the well at 0 units.  Are each of the boundary conditions stated in #3 of the Introduction satisfied?   Is it possible for the particle to exist outside of the well even if it's energy is less than the well depth?

Question

Does your conclusion regarding parity for the infinite square well still hold for the finite square well?

Question

What do the wave function and the energy levels look like if the energy of the particle is much greater than the well's depth.  Notice the behavior of the wave at the right boundary.  Describe the effect of the well on unbound wave functions as the energy is decreased.

Credits

Physlet problems authored by Dan Boye.