Physics – Quantum Physics
Scientific paper
2005-11-06
Journal of Physics A39, 11029-11051 (2006)
Physics
Quantum Physics
27 pages
Scientific paper
10.1088/0305-4470/39/35/006
A closed-form solution to the energy-based stochastic Schrodinger equation with a time-dependent coupling is obtained. The solution is algebraic in character, and is expressed directly in terms of independent random data. The data consist of (i) a random variable H which has the distribution P(H=E_i) = pi_i, where pi_i is the transition probability from the initial state to the Luders state with energy E_i; and (ii) an independent P-Brownian motion, where P is the physical probability measure associated with the dynamics of the reduction process. When the coupling is time-independent, it is known that state reduction occurs asymptotically--that is to say, over an infinite time horizon. In the case of a time-dependent coupling, we show that if the magnitude of the coupling decreases sufficiently rapidly, then the energy variance will be reduced under the dynamics, but the state need not reach an energy eigenstate. This situation corresponds to the case of a ``partial'' or ``incomplete'' measurement of the energy. We also construct an example of a model where the opposite situation prevails, in which complete state reduction is achieved after the passage of a finite period of time.
Brody Dorje C.
Constantinou Irene C.
Dear James D. C.
Hughston Lane. P.
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