Structure and properties of Hughston’s stochastic extension of the Schrödinger equation

Hughston has recently proposed a stochastic extension of the Schr\"odingerequation, expressed as a stochastic differential equation on projective Hilbertspace. We derive new projective Hilbert space identities, which we use to givea general proof that Hughston's equation leads to state vector collapse toenergy eigenstates, with collapse probabilities given by the quantum mechanicalprobabilities computed from the initial state. We discuss the relation ofHughston's equation to earlier work on norm-preserving stochastic equations,and show that Hughston's equation can be written as a manifestly unitarystochastic evolution equation for the pure state density matrix. We discuss thebehavior of systems constructed as direct products of independent subsystems,and briefly address the question of whether an energy-based approach, such asHughston's, suffices to give an objective interpretation of the measurementprocess in quantum mechanics.Comment: Plain Tex, no figure