# Stochastic Schroedinger equation from optimal observable evolution

Abstract : In this article, we consider a set of trial wave-functions denoted by |Q> and an associated set of operators A$_{alpha}$ which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form D = |Qa>, are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e. when k increases. The Monte-Carlo process might even become exact if the Hamiltonian H applied on the trial state can be written as a polynomial of A$_{alpha}$ The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions
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Journal articles

http://hal.in2p3.fr/in2p3-00171895
Contributor : Sandrine Guesnon <>
Submitted on : Thursday, September 13, 2007 - 2:27:43 PM
Last modification on : Monday, July 6, 2020 - 3:03:45 PM

### Citation

D. Lacroix. Stochastic Schroedinger equation from optimal observable evolution. Annals of Physics, Elsevier Masson, 2007, 322, pp.2055-2076. ⟨10.1016/j.aop.2006.09.003⟩. ⟨in2p3-00171895⟩

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