Here the \(\ket{m^j}\) are orthonormal states of the apparatus, \(\inpd{m^j}{m^k}=\delta_{jk}\), corresponding to different pointer positions, and the \(\ket{w^j}\) are normalized, \(\inp{w^j}=1\), but not necessarily orthogonal; it may be that \(\inpd{w^j}{w^k}\neq 0\) when \(j\neq k\). (In von Neumann’s original model the \(\ket{w^j}\) were the same as the \(\ket{s^j}\), and this is also possible, but is not being assumed, for the situation we are considering.) The transformation \eqref{eqn19} can be extended to a unitary transformation on \(\HC_s\otimes\HC_m\), because the orthogonality of the \(\ket{m^j}\) ensures that the states on the right side of \eqref{eqn19} are normalized and mutually orthogonal. Unitary time development leads to states