TY - JOUR

T1 - Structure and properties of Hughston's stochastic extension of the Schrödinger equation

AU - Adler, Stephen L.

AU - Horwitz, Lawrence P.

PY - 2000/5

Y1 - 2000/5

N2 - Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hubert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics.

AB - Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hubert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics.

UR - http://www.scopus.com/inward/record.url?scp=0034347307&partnerID=8YFLogxK

U2 - 10.1063/1.533255

DO - 10.1063/1.533255

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AN - SCOPUS:0034347307

SN - 0022-2488

VL - 41

SP - 2485

EP - 2499

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 5

ER -