Harmonic oscillator - wavefunctions I understand now how I can derive the lowest energy state $W_0 = \tfrac{1}{2}\hbar \omega$ of the quantum harmonic oscillator (HO) using the ladder operators. What is the easiest way to now derive possible wavefunctions - the ones with Hermite polynomials?
I need some guidance first and then I will come up with a bit more detailed questions.
 A: I think the easiest way to do this is to avoid solving differential equations to the greatest extent possible.  There is, in fact, a way to use ladder operators and only requires you to solve one, fairly easy differential equation;
First, we note that the ladder operator technique can be used to derive the entire spectrum of one-dimensional harmonic oscillator.
$$
  E_n = (n+\tfrac{1}{2})\hbar\omega
$$
The technique can also be used to show that the corresponding, properly normalized eigenvectors satisfy the following properties
$$
  a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle, \qquad a|0\rangle = 0
$$
The left-hand property shows that, once one has one of the eigenstates, every other eigenstate corresponding to a higher eigenvalue can be obtained by applying the raising operator.  In particular, if one knows the position basis representation of the ground state, then one can obtain the position basis representation of every other eigenstate by applying the position basis representation of the raising operator.
The right-hand property shows that the ground state is annihilated by the lowering operator.  Writing this condition in the position basis, one obtains a simple differential equation for the ground state wavefunction, and then, per the left-hand property, one generates all other wavefunctions. 
