Dynamics by white-noise Hamiltonians (original) (raw)

Abstract

A new class of random quantum-dynamical systems in continuous space is introduced and studied in some detail. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time-independent, and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time and arbitrary, but smooth in position and momentum. Exact expressions for the time evolution of both (mixed) states and observables averaged over randomness are obtained. The differences between the quantum and the classical behavior are clearly exhibited. As a special case it is shown that, if the deterministic part corresponds to a particle subjected to a constant magnetic field, the spatial variance of the averaged state grows diffusively for long times independent of the initial state. 05.40.+j, 05.60.+w, 72.70.+m The spatial spreading of a state under the free time evolution is a well-known and fundamental phenomenon in non-relativistic quantum and classical mechanics [1]. In order to make a quantitative statement, let σ 2 t denote the variance of the position at time t of a spinless point particle moving in continuous space. It is assumed that the particle was prepared initially in some state, which is normalized but not necessarily pure. For a free particle a simple calculation then shows that σ 2 t increases asymptotically for large t as σ 2 t ∼ t 2 ̺ 2 0 /m 2 . Here m > 0 is the mass of the particle and ̺ 2 0 is the variance of its momentum in the initial state. This relation holds both in the quantum and in the classical case. However, classical states, e.g. pure ones, may have a sharp momentum, that is, ̺ 2 0 = 0, whereas ̺ 2 0 > 0 for all quantum states, including the pure ones ("wave-packet spreading"). It is clear that σ 2 t can also be calculated exactly for a time evolution governed by a more general Hamiltonian being at most quadratic in momentum and position .

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