Occupation (or particle) number operator. Eigenvalues and eigenvectors.

Question

https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf

Help me please. I made a screen (below) from the article above and highlighted what I did not understand. Why is it true?

enter image description here

Answer 1

The operator $N$ works like this

$$ N \psi_{} = \psi_{} \tag{1} $$

Which you can read as: If I apply $N$ to a state, I get the label of state times the state (more technically, $\psi_{}$ is an eigenvector of $N$ with eigenvalue $$).

In your problem you have

$$ N(a^\dagger\psi_\nu) = (\nu + 1)(a^\dagger \psi_\nu) \tag{2} $$

If you compare this with (1) you will conclude that

$$ a^\dagger\psi_\nu \sim \psi_{\nu + 1} $$