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The Journal of Chemical Physics

Transmission function vs energy splitting in tunneling calculations. II. Computer simulation results

Ka-wai Lau, Jerome H. Weiner

Published Aug 15, 1985• 1 citation
Publisher

Abstract

An idealized model for the quantum tunneling of a particle in interaction with a heat bath is studied by computer simulation. The model may be motivated by consideration of a bound hydrogen–oxygen pair in a metal lattice; the H atom is treated quantum mechanically, the O atom is treated classically, and a double-well potential couples the two. In order to represent the O atom interactions with the rest of the lattice as heat bath, its motion is described by a classical Langevin equation. Computer simulation of the model shows that, with all other parameters held fixed, tunneling through low energy barriers is governed by energy splitting, while for high energy barriers the transmission approach agrees well with simulation results.

DOI
10.1063/1.449393

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Two methods have been widely employed for computing tunneling rates in a double-well potential, one based on a transmission function, the second on energy splitting. Semiclassical calculations (Brickmann and Zimmermann) show that the transmission method leads to lower rates than the splitting method. It is shown here without employing the semiclassical approximation that a more accurate relation between the energy splitting and the transmittion function may be obtained by using a decomposition of the stationary states into scattering states. This relation is then used to provide an analytical basis for the usual heuristic picture in which the particle oscillates with classical frequency in one well and has a probability of tunneling to the other well equal to the transmission function in each classical period of oscillation. It is concluded that the transmission method should give more meaningful results, particularly in situations where interactions of the system with a heat bath have significant effects in the tunneling times predicted by the splitting method. This is in accord with previous work in which both methods were used in an attempt to fit experimental data for hydrogen diffusion in niobium.

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An idealized model for the quantum tunneling of a particle in interaction with a heat bath is studied by computer simulation. The model may be motivated by consideration of a bound hydrogen–oxygen pair in a metal lattice; the H atom is treated quantum mechanically, the O atom is treated classically, and a double-well potential couples the two. In order to represent the O atom interactions with the rest of the lattice as heat bath, its motion is described by a classical Langevin equation. Computer simulation of the model shows that, with all other parameters held fixed, tunneling through low energy barriers is governed by energy splitting, while for high energy barriers the transmission approach agrees well with simulation results.
Two methods have been widely employed for computing tunneling rates in a double-well potential, one based on a transmission function, the second on energy splitting. Semiclassical calculations (Brickmann and Zimmermann) show that the transmission method leads to lower rates than the splitting method. It is shown here without employing the semiclassical approximation that a more accurate relation between the energy splitting and the transmittion function may be obtained by using a decomposition of the stationary states into scattering states. This relation is then used to provide an analytical basis for the usual heuristic picture in which the particle oscillates with classical frequency in one well and has a probability of tunneling to the other well equal to the transmission function in each classical period of oscillation. It is concluded that the transmission method should give more meaningful results, particularly in situations where interactions of the system with a heat bath have significant effects in the tunneling times predicted by the splitting method. This is in accord with previous work in which both methods were used in an attempt to fit experimental data for hydrogen diffusion in niobium.
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.
The tunneling effect is the most popular phenomenon of quantum physics and is present in modern physical theories. Still, the most important features of this effect are already present in toy models - low dimensional quantum mechanics with high potential barriers. Even these simple systems cannot be solved analytically and approximations have to be made. Tunneling is closely related to splitting of the ground state energy which is degenerate in the classical limit. In this thesis we use two methods to estimate the energy splitting in one dimensional quantum mechanics. In semi-classical approximation the splitting is calculated in the weak coupling limit. The other approach, Tamm-Dancoff method gives very precise numerical results for finite coupling. Comparing these two approaches we are able to confirm results of the semi-classical approximation and determine range of its applicability. Several potentials are considered: anharmonic double well, cosine potential and cosine potential in periodic space with finite number of minima. We also address a non-generic problem of anharmonic triple well potential.
A model for interstitial diffusion is studied in which the impurity atom is regarded as a Brownian particle constrained to move in a single direction and subject to a periodic potential. A quantum mechanical theory is developed for the rate at which atoms leave a given potential well. The theory makes use of an ensemble of minimum-uncertainty wave packets to describe thermal equilibrium in the well and computes the tunneling probability through the potential barrier on the basis of the Schr\"odinger-Langevin equation in order to incorporate the effect of an interacting heat bath. The result of the interactions is to reduce the departure rate at a given temperature level, but the qualitative nature of the rate expression remains unchanged and predicts Arrhenius plots with straight-line behavior at high-temperature levels and curvature due to tunneling at low-temperature levels. The dynamics of the wave packet after it leaves a well is studied by computer solution of the Schr\"odinger-Langevin equation and it is found that the persistence of motion decreases with increasing strength of interaction with the heat bath.
A real-time functional-integral method is used to derive an effective action that gives the transmission spectrum of a tunneling particle interacting with a bath of harmonic oscillators. The transmission spectum is expressed in terms of double functional integrals with respect to the coordinate of the particle which are evaluated by means of the stationary-phase approximation. The equations of motion for the stationary-phase trajectories are solved exactly for an arbitrary spectral density function of the bath, and the obtained solutions are used to find the transmission spectra for specific examples. For a bath with single frequency \ensuremath{\omega}, an analytic expression of the transmission spectrum is obtained which covers from the sudden tunneling (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1) to the adiabatic one (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\gg}1), where T0{\mathit{T}}_{0}T0​ is the time it would take a classical particle to traverse the inverted bare potential barrier. For a bath with an Ohmic spectrum, the differential tunneling conductance at low bias voltage V and for \ensuremath{\eta}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1 is found to obey a power law \ensuremath{\sim}(eVT0{\mathit{eVT}}_{0}eVT0​/\ensuremath{\Elzxh}${)}^{\mathrm{\ensuremath{\eta}}{\mathit{T}}_{0}{\mathit{S}}_{0}/2\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{\Elzxh}}}$, where \ensuremath{\eta} is the friction coefficient and S0{\mathit{S}}_{0}S0​ is the tunneling exponent in the absence of interaction. \textcopyright{} 1996 The American Physical Society.
The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.