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arXiv:1810.01318

Characteristic times for thermal wave packets in dissipative Bohmian mechanics: the parabolic repeller

S. V. Mousavi, Salvador Miret-Artés

Published Oct 2, 2018• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1810.01318

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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.
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.
Diffraction in time and early-arrival phenomena will be studied in the context of dissipative systems. This theoretical consideration is done by using the linear scaled wave equation proposed in the framework of Caldirola-Kanai theory for the continuous quantum-classical transition. Two simple but physically important and interesting examples are studied, sudden release from a shutter where transient behaviors (diffraction in time) are seen; and also transmission through a time-dependent parabolic barrier where early-arrival is seen. Calculations show that diffraction in time is a non-classical effect and temporary behaviors are gradually suppressed as the friction increases. Furthermore, early-arrival is seen even in the classical regime. This means that early-arrival is not a non-classical phenomenon. This behavior will be explained by means of scaled trajectories which are just Bohmian trajectories in the quantum regime.
Ignoring thermal fluctuations of the environment, taking into account only its dissipative effects, free propagation of a Gaussian wavepacket is studied in the framework of the linear Caldirola-Kanai (CK) equation and non-linear equations, the Schrodinger-Langevin (SL) equation, known as Kostin equation, and the Schuch-Chung-Hartmann (SCH) equation. By a Gaussian ansatz for the probability density one obtains two equations, one for the evolution of the center of the wave packet which is just the classical Langeving equation and one for the evolution of the width of the wavepacket. This last equation has different forms in different approaches having only an analytic solution in the CK and SCH frameworks. Computations show that for a given friction, the width of the wavepacket increases with time in all approaches. In a given time, it reduces with friction in both CK and SL approaches revealing localization effects of dissipation, while has opposite behavior in the SCH approach. Furthermore, energy expectation value and its time-derivative is computed and compared in all approaches. It is shown that the rate of energy is given by the expectation value of momentum field in the SL framework. Finally, transmission of a Gaussian wavepacket from a rectangular barrier is numerically studied in the context of CK and compared to the non-dissipative case. The transmission decreases considerably with the dissipation.
Usually tunneling is established after imposing some matching conditions on the (time-independent) wavefunction and its first derivative at the boundaries of a barrier. Here an alternative scheme is proposed to determine tunneling and estimate transmission probabilities in time-dependent problems, which takes advantage of the trajectory picture provided by Bohmian mechanics. From this theory, a general functional expression for the transmission probability in terms of the system initial state can be reached. This expression is used here to analyze tunneling properties and estimate transmissions in the case of initial Gaussian wave packets colliding with ramp-like barriers.
Within the framework of Bohmian mechanics dwell times find a straightforward formulation. The computation of associated probabilities and distributions however needs the explicit knowledge of a relevant sample of trajectories and therefore implies formidable numerical effort. Here, a trajectory free formulation for the average transmission and reflection dwell times within static spatial intervals [ a , b ] is given for one-dimensional scattering problems. This formulation reduces the computation time to less than 5% of the computation time by means of trajectory sampling.
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.