Okay I posted a question a few days ago about Luders Rule but didn't get any responses. I've studied this stuff in Hughes (The Structure and Interpretation of Quantum Mechanics) a bit more so I can ask a slightly different question. Hughes says you can create a "generalized probability function" that can apply to situations where a classical probability function can't. In particular, they apply to the "orthoalgebras" that describe quantum observables (where some observables are incompatible). When these generalized probability functions are expressed using density operators many wonderful things happen. In particular composite systems (like the 2-slit experiment) are represented by a new pure state, rather than a mixture of states. I.e. the 2-slit experiment with both slits open is a totally different pure state from the state with only one slit open. He then says that when you use the time-dependent Schrodinger equation on this pure state, you get "diffraction of the state function" which explains the interference pattern on the screen: its due to evolution of the state function as the state evolves in moving from the slit to the screen. This seems really neat, but I don't understand what the "generalized probability function" brings to the density operator/projector/Hilbert space formalism that makes all this possible. Can anybody explain what these wonderful generalized probability functions are that can do all this? Thanks.