Eigenvalues and Eigenstates of Number operator I have been working through a problem. It has asked me to determine the eigenstates and corresponding eigenvalues of the number operator in a quantum harmonic oscillator;
$$\hat{n}=\hat{a}_+\hat{a}_-$$
I have been looking for some literature on it but I can't seem to find anything! I know what solution I am expecting. I believe since the Hamiltonian and the number operator commute, then we can say that they must share the same family of eigenstates, which in the case of the Quantum harmonic oscillator come in the form of the Hermite Gauss Polynomials. My problem is where to really start. I'm unsure as to how this 2 term equation can blow up into the long expression for the eigenstates of the Hamiltonian operator in the quantum harmonic oscillator. I have also found some videos that go through the calculation, they find that the corresponding eigenvalues can be any positive integer. Any help would be greatly appreciated!
 A: Given that this is a homework problem, I'll just outline the general procedure that you need to follow: your argument is correct, since the Hamiltonian and the Number operator commute, there must be at least one simultaneous eigenbasis for the two operators. (There is no guarantee in general in such a case for every eigenbasis to be the same, of course the form of the Hamiltonian should give you a clue as to whether or not the "standard" energy eigenbasis would work.)
Suppose now you want to do this as generally as possible. Keep in mind that the eigenstates of the Harmonic Oscillator form a complete set. Let's call them $|n\rangle$. Since they do, any arbitrary state $|\psi\rangle$ can be written as a linear combination of these states.
In particular, a candidate for an eigenvector of $\hat{a}_{+}\hat{a}_{-}$ can also be written as $$|\psi\rangle = \sum_{n=0}^\infty c_n |n\rangle,$$
and your job is now to calculate the possible $c_n$s and $\lambda$s that would allow the following relation to hold:
$$\hat{a}_{+}\hat{a}_{-} |\psi\rangle = \lambda |\psi\rangle.$$
If you simply plug in the form of $|\psi\rangle$ given above, and remember that the different $|n\rangle$ states are linearly independent, you should be able to find that only certain values of $\lambda$ and $c_n$ are allowed. That should give you the states $\psi_k$ and eigenvalues $\lambda_k$ of the $\hat{N}$ operator. (You can verify your answer by keeping in mind that it's called the number operator for a reason!)
