These are two examples of using the quantum mechanics Physlet "Eignenvalue". These applets display the wavefunctions for the standard quantum problems of the infinite and finite square wells. Included are both displays of the wavefunctions and controls for the energy of the state being computed. A left-to-right shooting method is used to solve for the wavefunctions. For the finite square well the functional form for the shape of the well can be input.
Questions about the states and their energy and spatial dependence are included.
Type of Material:
Curricular Material (Lecture/Demo or Homework Assignment) (Scriptable)
Technical Requirements:
These are examples of the use of Physlets, scriptable building blocks for online physics demonstrations and other learning activities. For learning how to use Physlets skillfully, "Physlets: Teaching Physics with Interactive Curricular Material" by W. Christian and M. Belloni is recommended.
Identify Major Learning Goals:
Pictorial description of quantum states to develop "intuition" in students.
Demonstration of the importance of boundary conditions in solving quantum bound state problems.
Target Student Population:
Sophomore or Junior Modern Physics or Quantum Mechanics Classes
Prerequisite Knowledge or Skills:
Introduction to Quantum Mechanics. Could be used as supplemental material for more advanced classes.
Content Quality
Rating:
Strengths:
This is a very useful pictorial display of quantum wavefunctions. It gives a clear display of the wavefunctions of the square well. It is also an example of the shooting method for solution of differential equations. Users have a great deal of control over the applet, including the energy of the state being computed and the shape of the well for the finite square well example.
The wavefunctions can be saved in separate windows so that they can be compared. This will allow students to explore the physics of different states.
More examples of how to use this applet are available at http://webphysics.davidson.edu/Applets/java11_Archive.html under "Eigenvalue" in the table of contents. Some of these are quite interesting, although they are not a complete package with instructions and questions like the examples reviewed here.
Concerns:
The units used in these applets are potentially confusing. Students are asked questions to determine the units employed for the infinite square well (hbar2/2m = 1). If they don't understand this, it will be difficult for them to determine quantitative results from the graphs. There is no discussion of units for the finite square well. There is no indication of units on the graphs themselves.
Left clicking on the wavefunction graphs gives results of x, E, and y. These last two are confusing because it is uncertain what the E corresponds to (moving the cursor changes E?), and the values of y imply that the wavefunctions are not normalized to 1.
For the finite square well, it is unclear if the wavefunctions for different energies are normalized to the same relative value. This will distort the relative shapes of wavefunctions of different energies,
or wavefunctions for different size wells.
In the infinite well applet, the boundary conditions on the calculation are listed as Y(a) = Y(-a) = 0. It should be stated that in the simulation a = 1, as is done in the Physlets book for the finite square well. This probably should be stated directly in both cases on the web site.
The term "Principal" quantum number is used in these examples. This is nonstandard usage for 1D bound state problems where there is only one quantum number.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
Recommended Use(s) for Material (used for Effectiveness Review): Lecture/Demo for courses on modern physics or quantum mechanics, or Homework Assignment for a course on quantum mechanics. Most effective because of its interactivity.
Includes questions regarding the qualitative and quantitative form of quantum wavefunctions. These direct students to explore some of the physics behind quantum bound states.
The ability for students to change the energy of the state and the form of the potential is very powerful. This allows students to explore very carefully the physics of the square well without needing to deal with the transcendental equations one gets for the energies in standard solution approaches. This is particularly useful for lower level courses where students might not have the mathematical sophistication to solve the problem and understand the solution.
Concerns:
Use(s) Material is not Suited For: Tutorials (Description of the physics is not included.)
The author correctly admits that "students can be confused by the plotting of the wave function with the horizontal axis shifting to the energy eigenvalue." Plotting the wave functions and energy levels on separate graphs is pedagogically a sounder approach
Some comments on what a "shooting method" implies should be included. Many students,
and some instructors, will not know this. Some idea of this concept is needed to solve the Question 10.3.2a regarding proper boundary conditions. Also a mention that this problem can be solved analytically should be included.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
Easy to understand controls and input for displaying the wavefunctions. This is a very straightforward applet to use.
The instructions on how to use this applet are very good.
The ability to save and display different plots, and the instructions on how to do so, are a great benefit.
This example can be plugged, as-is, into a course on quantum mechanics. As is true with all Physlets, the flexibility of this applet allows a wide range of uses, with some work on the part of the instructor.
Concerns:
The buttons used for changing the energy of the state being computed could be labeled a little more clearly (rather than ">", ">>", "<", and "<<").
More information regarding the changing of the potential for the finite square well would be very useful. This is a very powerful feature of this applet and some examples would improve the usability. Students may attempt other potential shapes and become discouraged in their attempts without more direction.
Creative Commons:
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