Using ladder operators, I can find eigenstates $\psi_n$ with eigenenergies $$E_n=\hbar\omega\left(n+\frac{1}{2}\right). $$ In my textbook, ladder operators work like $$ a\psi_n = c_n \psi_{n-1}$$ $$ a^\dagger \psi_n = d_n \psi_{n+1} $$ where $c_n$ and $d_n$ are propotional constants.
But, how do I assure that the difference between eigenstates is just $\hbar\omega$ and there is no other eigenstates between them? In other words, how do I assure that ladder operators make the right next eigenstate?