The Hamiltonian of a two-level system is given by $$H=E_1|1\rangle\langle 1|+E_2|2\rangle\langle 2|$$ where both the energy eigenstates $|1\rangle$ and $|2\rangle$ are non-degenerate with $E_2>E_1$. This means that there is no other operator which commutes with the Hamiltonian $H$. Now consider such a 2-level system with $N$ atoms. Let at $t=0$, there is a population inversion so that all the $N$ atoms are in the excited state $|2\rangle$ and zero atoms in the ground state $|1\rangle$.

1. Does it mean that one implicitly defines a number operator and which has been measured? If yes, I don't know how to define it.

2. If no, how can I claim that there are $N$ atoms are found in the state $|2\rangle$ and $0$ atoms in $|1\rangle$?

The Hamiltonian of a two-level system is given by $$H=E_1|1\rangle\langle 1|+E_2|2\rangle\langle 2|$$ where both the energy eigenstates $|1\rangle$ and $|2\rangle$ are non-degenerate with $E_2>E_1$. Now consider such a 2-level system with $N$ atoms. Let at $t=0$, there is a population inversion so that all the $N$ atoms are in the excited state $|2\rangle$ and zero atoms in the ground state $|1\rangle$.

1. Does it mean that one implicitly defines a number operator and which has been measured? If yes, I don't know how to define it.

2. If no, how can I claim that there are $N$ atoms are found in the state $|2\rangle$ and $0$ atoms in $|1\rangle$?