The reader may have noticed that the particle states \(\ket{w^j}\) in \eqref{eqn19} played no role in our discussion of measurement in subsection 7.2. Von Neumann’s original (1932: sec. VI.3) presentation used \(\ket{w^j}=\ket{s^j}\), as he thought it important that a second measurement could confirm the first. This practice was followed in later textbooks, and has resulted in some confusion. The \(\ket{w^j}\) are in fact irrelevant when considering measurements, but they are useful when considering a preparation procedure in which macroscopic apparatus is used to produce a quantum system—again we shall speak of it as a particle—in some well-defined state. To see how this works let us again employ the model of subsection 7.1 with unitary development given by \eqref{eqn19}, but now introduce a family