Towards solids

Exercise 5

In this task we will try to illustrate what happens if many individual finite wells are connected together. The idea is that the results should reflect the difference between a single finite well and many finite wells representing one atoms to many atoms.
Open the file "Exercise 5 - towards solids.flow". A node diagram appears with the following content:

  • "Spatial dimension" - definition of x-axis.
  • "Number of wells" - a packed structure that defines a series of final wells (each with width 2) and with a distance between them defined by the number "separation". The five outputs correspond to 1, 2, 3, 4 or 5 wells. You can unpack the structure by clicking on the small arrow at the bottom of the field.
  • Two "energy plots". The upper one is programmed to display the energy levels in a single well, the bottom of five wells. You can move the "wires" if you want to see the example of 2, 3 or 4 wells.

Answering the following questions:

  1. How many bound states are there for a single well? How many are there for the five wells? This is most easily seen by changing "$N_{eigenstates}$" in "Energy plot". Start with the value 1 and then gradually increase the value.
  2. Note that the energies "come in groups" for the five wells. How many solutions are there in each group?
  3. What conditions are met when energies bundles into a single group?
  4. You can also look at the individual wave functions by clicking on $\{\psi_n\}$ in the "Energy plot". How do the wave functions next to the five wells compare to the case of a single well?
  5. What happens to the wave functions and energies of the individual groups if the distance between the wells change? You can adjust the distance to the left in the diagram by changing the "Separation between wells".
  6. Discuss what happens if there are not five wells, but a very large number - just as there are many atoms in a solid. If you have heard about about energy band gaps in solids, discuss the relation with them.