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In standard nonrelativistic quantum mechanics the expectation of the energyis a conserved quantity. It is possible to extend the dynamical law associatedwith the evolution of a quantum state consistently to include a nonlinearstochastic component, while respecting the conservation law. According to thedynamics thus obtained, referred to as the energy-based stochastic Schrodingerequation, an arbitrary initial state collapses spontaneously to one of theenergy eigenstates, thus describing the phenomenon of quantum state reduction.In this article, two such models are investigated: one that achieves statereduction in infinite time, and the other in finite time. The properties of theassociated energy expectation process and the energy variance process areworked out in detail. By use of a novel application of a nonlinear filteringmethod, closed-form solutions--algebraic in character and involving nointegration--are obtained for both these models. In each case, the solution isexpressed in terms of a random variable representing the terminal energy of thesystem, and an independent noise process. With these solutions at hand it ispossible to simulate explicitly the dynamics of the quantum states ofcomplicated physical systems.Comment: 50 page

Quantum noise and stochastic reduction