In our lecture today, we introduced two kinds of creation and annihilation operators.
I want to restrict myself to the antisymmetric case:
The first operator $a_k^{\dagger}$ creates a state $|k\rangle$ and the action is given by
$a_k^{\dagger} |1,...,N \rangle = \sqrt{N+1} |k,1,...,N \rangle$ , where $|1,...,N \rangle$ were antisymmetrized states(!) ( slater determinants).
Now a couple of minutes later, our lecturer switched to the notion of occupation numbers and introduced for $n_j$ being the occupation of state $k$
$a_k^{\dagger} |n_1,..,n_k,.,.. \rangle = (-1)^{N_{\beta}} \sqrt{ n_{\beta}+1} |n_1,...,n_{k}+1,.. \rangle,$ where $|n_1,..,n_k,.,.. \rangle$ is supposed to be an element of the Fock space and $N_{k}:= \sum_{i=1}^{k-1} n_i$.
Now my problem with this is, that both operators apparently create a new state k, but I don't see why they act so differently( the first one gets me a $\sqrt{N+1}$, whereas the second one does something completely different, although both of them just create a new state $k$.
I mean, I think both of them look somehow plausible, but somehow they seem to contradict each other.
If anything is unclear, please let me know.