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.


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