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  • Quantum properties of the infinite square well.
Topics:  Wave functions, square well potential, probability amplitude, energy eigenvalues, numerical solution to the Schroedinger equation.

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

Approximate completion time:  Under an hour.

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

Applet by Wolfgang Christian
1.  Because the potential energy of the regions outside the walls of the well are infinitely large, the particle is completely confined to the well and cannot penetrate the wall.   Therefore, what are the proper boundary conditions for this physical problem?

2. Choose an energy eigenvalue (say, E = 120.903) and an energy that is not an eigenvalue (say E = 100) and display the corresponding wave functions.  Given your answer to Question 1, how do the wave functions that correspond to each energy differ?   How does this compare to the case of a finite square well.  (See Quantum Physics Web Assignment No. 2.)

3.  (Question by Dan Boye)  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?


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?   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).

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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