Question

Consider a harmonic oscillator. What is the relationship between the number of nodes and the vibrational energy level, as characterized by the vibrational quantum number?

Answer 1

For one dimensonal harmonic oscillator, there is no degeneracy of the energy states.
And so, if we now classify the different energy levels labelled by the vibrational quantum number n, then, we mention a few points :

There is no node ( node means the zero of the wave function except the end points) in case for the ground state ( n = 0).
There is only one node in the case for the first excited state (n = 1). And similarly,
  There are two nodes in the case for the 2nd excited state ( n = 2).
So, we generalize the statement and state as follows :
  The number of nodes in the n th state is n. So, the number of nodes of the state is equal to the vibrational quantum number of that quantum state.