This week's homework is difficult, I grant you that, but not without reason.
*The first two problems are really our first stab at calculating the properties of real, everyday, useful materials (not that H is not useful or interesting, I guess). The vibrational frequency you'll calculate for KCl matches experiments amazingly well, in spite of the simple model potential used.
* The variational principle is something you will come across again, probably in mechanics (PH301/2) if not quantum. It is more or less a powerful way to come up with a best guess solution to a problem without actually solving it, and therefore powerful. There exist even more powerful methods commonly used in Chemistry and Physics for calculating the electronic properties of materials (e.g., Hartree-Fock, Density Functional Theory), but they are far more difficult. Should you encounter them, you're likely to be thrown in the deep end; the point of problems 3&4 is to give you a taste of how to handle systems which cannot be treated exactly without all the mathematical baggage that can obscure the essential simplicity of the method.
* Coupled oscillators can be used to explain a really ridiculous number of phenomena. In Thursday's lecture, we'll used a coupled oscillator model to (more simply) re-derive all of what we've learned of bonding, and cast it in a form that looks suspiciously like masses & springs or coupled LC oscillators. With this new approach, we'll be able to extend our analysis to the case of periodic solids (like semiconductor crystals, leading us to transistors and such). We'll be able to explain why some stuff is electrically conducting and other stuff is not, and why real materials behave the way you do. Problem 5 is meant to get you thinking about how coupled oscillators work as a preface to that lecture. It also gives you some hints on how one can spectroscopically identify different molecules (look for radiation emission/absorption matching the vibrational frequencies) or when molecules are adsorbed on a surface, e.g., in catalysis (new vibrational modes show up compared to the original molecule).
So, in short (if it isn't too late for that), think of this problem set as a preview of what's to come - both how we'll figure out how to calculate the properties of real materials, and what you're going to be up against in later courses. Most of what we've done the last month or two has been leading up to this.
