Consider the example of a one-dimensional quantum harmonic oscillator. As is well known, its energy \(E\) can take on only discrete values \((n+1/2)\hbar\omega\), where \(n=0, 1,\ldots\) is any nonnegative integer and \(\omega\) is its angular frequency, as in \eqref{eqn1}. We denote the corresponding normalized ket by \(\ket{n}\). The projector \([n]=\dya{n}\) then represents the property that the energy is equal to \((n+1/2)\hbar\omega\). The property that the energy is less than or equal to \((N+1/2)\hbar\omega\) for some integer \(N\) is given by the projector