Here is the HOMEWORK for this week

2) There are many ways to get this one. The easiest might require taking a look at problem 5a from last week, where you proved that <T> = <V> for a harmonic oscillator in an eigenfunction of energy, which should help you quickly figure out what <x^2> and <p^2> are. (Symmetry should help you figure out <x> and <p>) . There's very little calculational work you have to do for this problem!

3) In class (and in Gas as well) we went through the derivation of the energies of the harmonic oscillator (i.e, of the eigenvalues of H) without solving any differential equations of any kind, just thinking about commutators, and ladder operators. In this problem, you should do pretty much exactly the same. The operator is now N, rather than H, but they are in fact extremely similar. The object is for you to go through the derivation of eigenvalues, following the logic of how ladder operators affect the eigenvalues (why must there be a lowest state for N? Can you quickly show that N can't have a negative expectation value?), etc...