• Quantum properties of the finite square well.

Pre-requisite skills:  An understanding of basic quantum physics and the meaning of the wave function and eigenvalue problem.

Approximate completion time:  Under an hour.

Provide sufficient detail to verify that the assignment was completed in a meaningful manner.

Applet by Wolfgang Christian

2. Choose n = 5 and examine the corresponding wave function.  Now move the energy away from teh energy eigenvalue by pressing the ">>" button.

(a)  What happens to the wave function?

(b)  Does the wave function obey the boundary conditions from Question 1?

3.  Set E = 310 and, using the ">>" button, increase the energy until the next eigenvalue is found. 

(a)  What is this value?  

(b)  How many nodes are in the wave function?

(c)  What is the relationship between the number of nodes and the principal quantum number n for this problem.

4.  (Question by Dan Boye)  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 six energy levels?  What general conclusion can you draw regarding the [principal] quantum number and the parity for an arbitrary energy level?

5.   The points where the wave function is 0 (not including the end points) are called nodes.  The principal quantum number n is related to the number of nodes in the wave function.   Choose n = 1 in the text box and examine the resulting wave function.  Now choose another n and do the same.   What is the relationship between the number of nodes and the principal quantum number in this problem?

6.   This applet does not display the probability, but rather the probability amplitude, of finding the particle at a particular position.   What is the difference between the two terms?   Which corresponds to a more physical property?   Sketch both the for fourth-excited state (that is, the wave function corresponding to    n = 5).

7. Using your experience with this applet, explain why it is stated that imposing physical boundary conditions on a quantum mechanical system causes the resulting energy spectrum to be quantized (that is, discrete).

8.  The width of the well can be changed by replacing the 0.5 in the potential energy function to a larger value.   Let us call this value W.   Change W = 0.5 to W = 0.7 and count the number of bound states allowed by the resulting well.  Do the same for W = .3, and state the relationship between the width of the well and the number of allowed states.   Why is it said that narrowing a potential energy well "squeezes bound states into the continuum"?  (The continuum is that region above the top of the well.)

9. The number 1000 in the potential energy function corresponds to the height of the well.  Using similar analysis, how does the height of the well correspond to the number of allowed bound states for a given width? 

Helpful Resources

1. Vectors and Motion by Tom Henderson.
2. The Physics Hypertextbook by Glenn Elert (see Mechanics)
3. Physics E-Book - Projectile Motion by Fred Gram (formerly, Zogrseb, of the planet Ktoobirzp).
4. Book of Phyz - Motion by Dean Baird

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