Time-Dependent Gaussian Solution for the Kostin Equation Around Classical Trajectories

Abstract

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.

The Schrödinger–Langevin equation with and without thermal fluctuations

“…Within the SLE, we only need to assume that the noise operator can be taken as a commutating c-number. The latter assumption was actually already implied in Kostin's derivation of the SL random potential [13] and does not lead to a violation of the Heisenberg relations [31,32,33].…”

“…Quite generally, r x = r p and the p n generated from G are more involved than the simple power law p n ∝ c −n found in equation (32) for the white noise case. As a consequence, deviations from usual Boltzmann distributions are expected for the p n .…”

“…The evolution of a Gaussian wave packet under the SLE has been extensively discussed by several authors (see [32] and references therein). Here we derive anew the essential results for our study, concentrating on the asymptotic shape of the solution.…”

“…A nonlinear friction can be obtained by extending Kostin derivation to a nonlinear coupling [28,29]. The SLE exhibits interesting properties: unitarity is preserved at all times [24], the uncertainty principle is always satisfied 2 [31,32,33] and the superposition principle is violated due to the nonlinearities (which might not be a problem per se for dissipative equations [34,35]). Thanks to its straightforward formulation -in principle only two "classical" parameters need to be known: the friction coefficient A and the bath temperature T bath -and its numerical simplicity, the SLE can be considered as a solid candidate for effective description of open quantum systems hardly accessible to quantum master equations [14,36].…”

“…The study of its solutions without stochastic term has been carried out in many specific cases, either analytically [18,31,32,41,42,43,44,45,46] or numerically [24,33,41,47,48,49,50,51]. Along these analysis, it has been advocated that the stationary eigenstates of H 0 are also stationary states of the equation [18,43].…”