# Is occupation number of a certain quantum state an observable? [closed]

I know that the number operators are projective. Can I use the number operator for measurement if it is projective? Can I use it if I want to measure the number of particles (fermions) in a certain quantum state?

You can (in principle) measure any Hermitian operator $$\hat{O}$$, which, by the spectral theorem, can be written as a linear combination of projections:
$$\hat{O} = \sum_i \lambda_i \hat{P_i},$$
where $$\hat{P_i}$$ is the projection onto the eigenspace of $$\hat{O}$$ with eigenvalue $$\lambda_i$$.
Since projections are Hermitian, the above applies to them too. The only notable thing with projections is that their only non-zero eigenvalue is $$\lambda = 1$$, meaning the above sum will contain only one term.
To answer your question, given a number operator $$\hat{n_i}$$ associated to a single particle basis function $$\phi_i$$, measurement of $$\hat{n_i}$$ will yield a value $$\lambda \in \{0, 1\}$$. Given this value, the measurement will project the many-body wavefunction to the $$\hat{n_i} = \lambda$$ eigenspace. When $$\lambda = 1$$, this is the space in which $$\phi_i$$ is occupied with a single fermion. When $$\lambda = 0$$, this is the space in which $$\phi_i$$ is unoccupied.