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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?

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1 Answer 1

All infinite dimensional Hilbert spaces which have orthonormal bases which are countable are indeed isomorphic. (Obviously finite dimensional ones aren't!).

When people talk about "constructing spaces of states", they tend to be talking about constructing explicit representations for the states and operators, rather than constructing the abstract space of states.

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if there is no problems with the fact, that I get a complete set of commuting observables always is formed by only one operator N? –  xxxxx Dec 16 '12 at 13:44
    
I think that both in the "normal" harmonic oscillator case and in your modified harmonic oscillator, the number operator eigenvalues specifies all there is to be known about the state. Once you know how many excitations there are, everything can be derived from that. In more complicated systems, like, for example, the H atom in 3 dimensions, you can't necessarily do it with one observable - e.g. energy isn't enough, you also need to specify total angular momentum and (say) the z component. –  twistor59 Dec 16 '12 at 14:24
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