   where the right hand side means that \(a_{r}^{\dagger}(\mathbf{k})\) operates \(n_r (\mathbf{k})\) times on \(|0\rangle\), the state vector of the vacuum with no photons present.  The interpretation of these results is parallel to the one of the harmonic oscillator. \(a_{r}^{\dagger}(\mathbf{k})\) is interpreted as the creation operator of a photon with momentum \(\hslash \mathbf{k}\) and energy \(\hslash \omega_{\mathbf{k}}\) (and a polarisation which depends on \(r\) and \(\mathbf{k})\).  That is, equation (5.4) can be understood in the following way. One gets a state with \(n_r (\mathbf{k})\) photons of momentum \(\hslash \mathbf{k}\) and energy \(\hslash \omega_{\mathbf{k}}\) when the creation operator \(a_{r}^{\dagger}(\mathbf{k})\) operates \(n_r (\mathbf{k})\) times on the vacuum state \(|0\rangle\). Accordingly, \(N_r (\mathbf{k})\) is called the number operator and \(n_r (\mathbf{k})\) the ‘occupation number’ of the mode that is specified by \(\mathbf{k}\) and \(r\), i.e., this mode is occupied by \(n_r (\mathbf{k})\) photons. Note that Pauli’s exclusion principle is not violated since it only applies to fermions and not to bosons like photons. The corresponding interpretation for the annihilation operator \(a_r (\mathbf{k})\) is parallel: When it operates on a state with a given number of photons this number is lowered by one.