State Vector Collapse Probabilities and Separability of Independent Systems in Hughston's Stochastic Extension of the Schr\"odinger Equation

We give a general proof that Hughston's stochastic extension of the Schr\"odinger equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We also show that for a system composed of independent subsystems, Hughston's equation separates into similar independent equations for the each of the subsystems, correlated only through the common Wiener process that drives the state reduction.