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arXiv:1306.6607 · Annals of Physics

Dissipative Bohmian mechanics within the Caldirola–Kanai framework: A trajectory analysis of wave-packet dynamics in viscid media

Ángel S. Sanz, R. Martínez-Casado, H.C. Peñate-Rodríguez, Germán A. Rojas-Lorenzo, Salvador Miret-Artés

Published May 2, 2014• 28 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2014.04.016

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Classical viscid media are quite common in our everyday life. However, we are not used to find such media in quantum mechanics and much less to analyze their effects on the dynamics of quantum systems. Here we provide a Bohmian analysis of the friction dynamics led by hypothetical quantum viscid media, which will be simulated by means of the so-called Caldirola-Kanai Hamiltonian. As it is shown, this causal model enables a better understanding of phenomena, such as quantum localization (“quantum freezing”) or the quantum-to-classical transition. We have analyzed these phenomena in a series of paradigmatic situations of quantum mechanics, such as free wave-packet evolution, motion in a linear potential, damped harmonic oscillator and interference dynamics.
A nonlinear quantum–classical transition wave equation is proposed for dissipative systems within the Caldirola-Kanai model. Equivalence of this transition equation to a scaled Schrödinger equation is proved. The dissipative dynamics is then studied in terms of what we call scaled trajectories following the standard procedure used in Bohmian mechanics. These trajectories depend on a continuous parameter allowing us a smooth transition from Bohmian to classical trajectories. Arrival times and actual momentum distribution functions are also analyzed. The propagation of a Gaussian wave packet in a viscid medium under the presence of constant, linear and harmonic potentials is studied. The gradual decoherence process and localization are easily visualized and understood within this theoretical framework.
Thermal wave packets are used to analyze transmission probabilities and characteristic times through a parabolic repeller within the dissipative Bohmian mechanics. Thermal arrival, dwelling, transmission and reflection times are defined and calculated by using a Maxwell-Boltzmann distribution for the initial velocities of the incident particles. The dissipation is considered within the Caldirola-Kanai and Kostin approaches where a linear and nonlinear framework is used, respectively. The initial parameters are chosen to have only a dissipative tunnelling dynamics at zero temperature; at any nonzero temperature, transmission proceeds not only via tunnelling.