Imagine I have an observer $\mathcal O$, a quantum system $\mathcal S$ with Hilbert space $V_{\mathcal S}$, a Hamiltonian $H$, a self-adjoint operator $A$ acting on $V_{\mathcal S}$. The system is in the (normalized) state $|\psi_0\rangle$ at time $t=0$, and the observer measures the system at a ...