Potential Well

How do the energy levels in a finite well compare to the energy levels in a infinite well?

1. How does the energy of a quantum trapped in a well (1-D box) change if you shrink the box by a factor of 5? - Give a quantitative answer.
2. Sketch the first 4 wavefunctions for an infinitely deep potential well
   
   

3. Write a general expression for the wavefunction for quantons in an infintely deep well and show that they satisfy the Schroedinger equation. (Hint V = 0 inside the box).
4. What is meant by the phrase "Classically Forbidden Region"?
5. Suppose that the ground state energy for a quanton in an infinite well is 10 units. What is the energy of the quanton when it is in the n = 5 state?
6. How is the energy of the n-th energy level related to the ground state (n = 1) level? Express this as a simple mathematical relationship.
7. Compare what you have just done with the following two scenarios:
   1. A quanton in a shallow well (run applet)
   2. A quanton in a deep well (run applet)
8. How is the ground state energy and the wavefunction different between the 3 scenarios? Does the simple relation that you derived in part 5 still hold? Discuss this.
   
   
   

Simple Harmonic Oscillator

1. Investigate the case in which the potential energy is described by . Show that this can be re-written as when we use the usual relationship between k,m and w for a classical spring oscillator . Write the Schroedinger equation for this case.
2. Show that the wavefunction solves the Schroedinger equation. Why is it reasonable to assume that this is the ground sate solution? (Hint - look at the finite well and compare the number of "humps" in the wavefunction with the energy level quantum number n.
3. Show that the ground state energy is given by the expression .
4. Go to SHO applet and inspect the energy levels for the Simple Harmonic Oscillator. Write an expression for E_n.

What to Hand In and When

By Monday's class please hand in your answers to the questions from both parts of this studio class.