The plot on the left shows you the numerical solution to the Schrödinger Equation for a particle in a box of width W at the energy E.

The left edge of the box coincides with the border of the plot, the right edge is given by the gray vertical bar on the right. The red line indicates the chosen energy level E, the thick black line the numerical solution to the Schrödinger equation for this system at the selected energy: The wavefunction of the particle in a box of width W at energy E.

The initial choice for the energy does not give a physically reasonable solution: The wavefunction grows to infinity at the right end of the plot. As we know, the square of the wavefunction is proportional to the probability to find a particle at this location, which means that for this particular solution the probability would grow to infinity - which is impossible.

Furthermore we know that the probability o find the particle outside of the box (to the right of the gray bar) is zero, after all it's supposed to be in the box...

1. Allowed Energy Levels
a. Vary the energy slider to find an energy where you do not encounter this problem of the wavefunction growing to infinity, where the wavefunction (and hence the probability) will actually stay close to zero beyond the box, a so called "allowed" energy level, corresponding to a physically reasonable solution, one that can actually represent the wavefunction of a real particle.
Hint: Click on the slider once with the mouse and then use the right and left cursor keys for mini-steps.

b. Vary the slider through the entire range of energies to find all the allowed energy levels, those that lead to physically reasonable solutions. Note the energies E and the number of nodes in the wavefunction.

c. You have just discovered the quantization of energy for a particle in a box. Where does this quantization come from?

d. Index your solutions, starting with n=1 for the lowest energy solution, counting up and prepare a table listing your index n, the number of nodes in the wavefunction, and the energy E. Your index is the quantum number for a particle in a box. The first state (n=1) is called the ground state, all other states are excited states.

e. Find a relationship between the quantum number n and the number of nodes as well as the energy E.

2. Zero-Point Energy
a. Is there a physically reasonable solution (allowed energy level) for E=0?

b. Use the uncertainty principle to show that this is not possible, that the particle can not be confined within the box and simultaneously have a total energy of E=0.

3. Width of the box
Adjust the energy slider to the third excited state (n=4) for W=1.

a. Now reduce the width W of the box a little. Does the same energy E still correspond to an allowed energy level?

b. Without changing E, reduce the width until the same energy corresponds to an allowed energy level again. Note the width W.

c. What quantum number does this energy level correspond to at this width?

d. Reduce the width further until you find two more allowed energy levels and their quantum numbers. List n, E, and W in a table.

e. Find the relationship between n, E, and W. Show that the relationship reduces to the relationship from 1.e. for W=1.

4. Absorption Spectrum
The transition of an electron between two levels correspond to the absorption of a photon, the energy difference between the two levels determines the absorption frequency.

a. Assuming an electron is trapped in your box, find the absorption frequency from the ground state to the first excited state (in energy units) for the full size box (W=1)

b. Change the width W a few times and repeat your simulation from 4.a. How does the absorption energy (frequency) shift?

c. How about the absorption wavelength? If the electron absorbs green light in a box of size W=1, what color will it absorb for W=0.75?