Number of operators in a complete set of commuting observables I just finished my first course on quantum physics and I have a question about the number of operators in complete set of commuting observables. 
For example, in the case of the hydrogen atom you can use the three observables $\hat{H}$, $\hat{L}^2$ and $\hat{L}_z$ and their corresponding quantum numbers to specify each eigenstate $|n,l,m\rangle$. Alternatively, you could use the three components of position $\hat{x}$, $\hat{y}$, $\hat{y}$ or momentum $\hat{p_x}$, $\hat{p_y}$, $\hat{p_z}$ to uniquely specify the eigenstates (via their wave functions). No matter which of these sets of observables you use, the number of commuting operators is the same, in this case three. I've also noticed this in other problems. Is this a general rule?
What I mean by this, is the number of operators in a complete set of commuting observables fixed for a given problem? Would this then imply that this number always corresponds to the number of degrees of freedom of the problem? 
If not, are there physical examples where it is possible to reduce the number of observables needed to completely specify the eigenstates of a system?
 A: Mathematically speaking, in an $N$-dimensional Hilbert space we can always construct $N$ linearly independent commuting observables. For example, let $|n\rangle$ be a basis of the Hilbert space and define
$$ A_k |n\rangle = \begin{cases} |n\rangle & n \leq k \\ 0 & n > k \end{cases} $$
for $1 \leq k \leq N$. (In particular, if the Hilbert space is infinite dimensional, one can construct an infinite number of independent commuting observables.)
In practice, we are usually interested in "physically interesting" observables such as the Hamiltonian. If we consider a set of commuting observables that includes the Hamiltonian, then all of these observables correspond to conserved quantities. In a system with an $n$-dimensional configuration space, there are at most $n$ independent conserved quantities. The systems studied in introductory lectures are often integrable, meaning that there are exactly $n$ conserved quantities. Which corresponds to your observation, that we often end up considering sets of 3 commuting observables.
