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Physical Review A

Schrödinger-Langevin theory for quantum tunneling through a barrier

B. J. McCoy, Ruben G. Carbonell

Published Jan 1, 1984• 5 citations
Publisher

Abstract

We have used the Kanai Hamiltonian, for a quantum-mechanical particle experiencing frictional interactions with its environment, to calculate the influence of dissipation in the surroundings on quantum-mechanical tunneling. We find that for the case of a rectangular potential-energy barrier, the dissipation can either increase or decrease the transmission coefficient, depending on the magnitude of the barrier height and width.

DOI
10.1103/physreva.29.399

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We have used the Kanai Hamiltonian, for a quantum-mechanical particle experiencing frictional interactions with its environment, to calculate the influence of dissipation in the surroundings on quantum-mechanical tunneling. We find that for the case of a rectangular potential-energy barrier, the dissipation can either increase or decrease the transmission coefficient, depending on the magnitude of the barrier height and width.
Using the phenomenological quantum friction models introduced by P. Caldirola [Nuovo Cimento 18, 393 (1941)] and E. Kanai [Prog. Theor. Phys. 3, 440 (1948)], M. D. Kostin [J. Chem. Phys. 57, 3589 (1972)], and K. Albrecht [Phys. Lett. B 56, 127 (1975)], we study quantum tunneling of a one-dimensional potential in the presence of energy dissipation. To this end, we calculate the tunneling probability using a time-dependent wave-packet method. The friction reduces the tunneling probability. We show that the three models provide similar penetrabilities to each other, among which the Caldirola-Kanai model requires the least numerical effort. We also discuss the effect of energy dissipation on quantum tunneling in terms of barrier distributions.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
A quantum system which can tunnel, at T=0T=0T=0, out of a metastable state and whose interaction with its environment is adequately described in the classically accessible region by a phenomenological friction coefficient $\ensuremath{\eta}$, is considered. By only assuming that the environment response is linear, it is found that dissipation multiplies the tunneling probability by the factor $\mathrm{exp}[\ensuremath{-}\frac{A\ensuremath{\eta}{(\ensuremath{\Delta}q)}^{2}}{\ensuremath{\hbar}}]$, where $\ensuremath{\Delta}q$ is the "distance under the barrier" and AAA is a numerical factor which is generally of order unity.
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.
The influence of a dissipative environment on scattering of a particle by a barrier is investigated by using the recently introduced bohmian mechanics with complex action [J. Chem. Phys. 125, 231103 (2006)]. An extension of this complex trajectory based formalism to include the interaction of the tunneling particle with an environment of harmonic oscillators with a continuous spectral density and at a certain finite temperature allows us to calculate transmission probabilities beyond the weak system bath coupling regime. The results display an increasing tunneling probability for energies below the barrier and a decreased transmission above the barrier due to the coupling. Furthermore, we demonstrate that solutions of a markovian master equation fail to do so in general.