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?

This post imported from StackExchange Physics at 2014-03-22 17:24 (UCT), posted by SE-user xxxxx