I want to learn how to construct spaces of quantum states of systems.
As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is the Hamiltonian of the harmonic oscillator with the quadratic term: $\hat{H}=\hat{H}_{0}+\hat{H}_{1}$, where
$\hat{H}_{0}=\hbar\omega\left(\hat{a}^{\dagger}\hat{a}+1/2\right)$, $\hat{H}_{1}=\dot{\imath}\gamma\left(\hat{a}^{\dagger}\right)^{2}-\dot{\imath}\gamma\left(\hat{a}\right)^{2}$; $\hat{a}$, $\hat{a}^{\dagger}$-ladder operators, $\gamma$-real parameter
For this purpose, we should define a complete set of commuting observables (CSCO).
As for the harmonic oscillator, we can define a "number" operator $N=\hat{a}^{\dagger}\hat{a}$.
We can prove the following statement:
Let be $a$ and $a^{\dagger}$ Hermitian conjugated operators and $\left[a,a^{\dagger}\right]=1$. Define operator $N=aa^{\dagger}$. Then we can prove that $\left[N,a^{p}\right]=-pa^{p}, \left[N,a^{\dagger p}\right]=pa^{\dagger p}$ and that the only algebraic functions of $a$ and $a^{\dagger}$, which commute with $N$, are the functions of $N$. (For example, see Messiah, Quantum Mechanics, exercises after chapter $12$)
Using this statement, we conclude(am i right?) that operator $N$ forms a CSCO. So, sequence of eigenvectors of operator $N$ forms the basis of the space of states. So, I've come to the conclusion that the space of states of the described system is the same as the space of states of harmonic oscillator.
But operators $a$, $a^{\dagger}$ can always be determined (as i think), so this arguments will be valid, so I've come to the conclusion, that spaces of states of all systems will be the same. After that I realized that I am mistaken.
Would you be so kind to explain where is a mistake in the arguments above? And can you give some references/articles/books where i can read some additional information about constructing spaces of states for different systems?
