# 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.

• it's a standard textbook derivation, see for example amazon.com/Quantum-Mechanics-2-vol-set/dp/0471569526 first volume chapter 5 – Ikiperu May 28 '13 at 11:55
• There is no "easiest way", there is just a standard way. But see Griffiths, Introduction to Quantum Mechanics for an intuitive, explanatory approach. – Man May 28 '13 at 12:26
• I am not sure that it is the easiest one, but the most common method is to solve corresponding Schrodinger equation in the coordinate representation. – freude May 28 '13 at 15:27
• I ll check Griffith for start – 71GA May 28 '13 at 17:18

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.