Finite Square-Well Potential

Although the finite square-well potential problem is more realistic, 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.

In this part of the lab you will see how the shooting method works.  Right-click and drag in the energy spectrum and you should see 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 = +) = 0.  A physically acceptable boundary condition is assumed at the left side of the graph and the solution is determined in the direction of increasing x. 

Exercises

  1. 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?
  2. 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 in Exercise 1?  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?
  3. 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 #2 of Section 1 met?   Is it possible for the particle to exist outside of the well even if it's energy is less than the well depth?
  4. Is it possible to find a combination of height and width such that there are no bound states?
  5. Does your conclusion regarding parity for the infinite square well still hold for the finite square well?
  6. 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.

Remember:
Left-click in the graph for graph coordinates.
Right-click in the graph to take a snapshot of the current graph.
Right-click drag the mouse inside the energy spectrum to change energy of particle.