Let us apply these ideas to a specific example, that of the spin degree of freedom of a spin-half particle, such as an electron or a proton, or a silver atom in its electronic ground state. The operator \(S_z\) for the \(z\) component of spin angular momentum acts on a 2-dimensional Hilbert space and has eigenvalues \(+1/2\) and \(-1/2\) in units of \(\hbar\). If \(\ket{z^+}\) and \(\ket{z^-}\) are the corresponding eigenkets, the projectors corresponding to \(S_z=\pm 1/2\) are \([z^+] =\dya{z^+}\) and \([z^-] =\dya{z^-}\). The product of these projectors in either order is the zero operator \(0\), which is the property that is always false, the quantum counterpart of the empty subset of a classical phase space. Their sum \([z^+] + [z^-] =I\) is the identity operator, the property that is always true. In physical terms, \(S_z\) is either positive or negative: it cannot be both (\([z^+]\cdot [z^-] =0\)), and it must be one or the other (\([z^+] + [z^-] =I\)). This explains the outcome of the Stern-Gerlach experiment.