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Australian Journal of Physics

Tunnelling of a Molecule

Peter Jarvis, Daniel Peter Bulte

Published Dec 21, 1998• 1 citation
Publisher

Abstract

A quantum-mechanical description of tunnelling is presented for a one-dimensional system with internal oscillator degrees of freedom.The 'charged diatomic molecule' is frustrated on encountering a barrier potential by its centre of charge not being coincident with its centre of mass, resulting in transitions amongst internal states.In an adiabatic limit, the tunnelling of semiclassical coherent-like oscillator states is shown to exhibit the Hartman and Büttiker-Landauer times t H and t BL , with the time dependence of the coherent state parameter for the tunnelled state given by α(t) = α e -iω(t+∆t) , ∆t = t H -it BL .A perturbation formalism is developed, whereby the exact transfer matrix can be expanded to any desired accuracy in a suitable limit.An 'intrinsic' time, based on the oscillator transition rate during tunnelling, transmission or reflection, is introduced.In simple situations the resulting intrinsic tunnelling time is shown to vanish to lowest order.In the general case a particular (nonzero) parametrisation is inferred, and its properties discussed in comparison with the literature on tunnelling times for both wavepackets and internal clocks.

DOI
10.1071/p98011

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A quantum-mechanical description of tunnelling is presented for a one-dimensional system with internal oscillator degrees of freedom.The 'charged diatomic molecule' is frustrated on encountering a barrier potential by its centre of charge not being coincident with its centre of mass, resulting in transitions amongst internal states.In an adiabatic limit, the tunnelling of semiclassical coherent-like oscillator states is shown to exhibit the Hartman and Büttiker-Landauer times t H and t BL , with the time dependence of the coherent state parameter for the tunnelled state given by α(t) = α e -iω(t+∆t) , ∆t = t H -it BL .A perturbation formalism is developed, whereby the exact transfer matrix can be expanded to any desired accuracy in a suitable limit.An 'intrinsic' time, based on the oscillator transition rate during tunnelling, transmission or reflection, is introduced.In simple situations the resulting intrinsic tunnelling time is shown to vanish to lowest order.In the general case a particular (nonzero) parametrisation is inferred, and its properties discussed in comparison with the literature on tunnelling times for both wavepackets and internal clocks.
In this article, we present the results of studies on the quantum mechanical tunneling and reflection of a diatomic, homonuclear molecule with a single bound state incident upon a potential barrier. In the first study, we investigate the tunneling of a molecule using a time-dependent formulation. The molecular wave function is modeled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration. It is found that a molecule may transition between the bound state and an unbound state numerous times during the process of reflection from or transmission past the barrier. It is also found that, in addition to reflecting and transmitting, the molecule may also temporarily straddle the potential barrier in an unbound state. In the second study, we consider the case of a molecule incident in the bound state upon a step potential with energy less than the step. We show that in the limit where the binding energy e0{e}_{0}e0​ approaches zero and the step potential V0{V}_{0}V0​ goes to infinity, the molecule cannot remain in a bound state if the center of mass gets closer to the step than an arbitrarily large distance x0{x}_{0}x0​ which increases as the magnitude of e0{e}_{0}e0​ decreases, as V0{V}_{0}V0​ increases, or both. We also show that, for ${e}_{0}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ and ${V}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$, if the molecule is incident in the bound state, it is reflected in the bound state with probability equal to unity, when the center of mass reaches the reflection distance x0{x}_{0}x0​. We verify that the unbound states exhibit the expected physical behavior. We discuss some surprising results. Connections between our results and investigations done in cold atoms, excitons, Cooper pairs, and Rydberg atoms are discussed.
The quantum tunneling rate of a macroscopic particle in one-dimensional potentials subject to oscillating external fields is considered using the complex time path integral method and the time-dependent Schrödinger equation. A simple calculation scheme for obtaining the tunneling rate within quasi-classical accuracy is described and applied to two systems. The one is the tunneling motion through an inverted parabola potential and the other is the escaping motion from a cubic potential well. Both systems are subject to small oscillating fields. It is found that both of the rates are exponentially enhanced due to the imaginary time motion of the tunneling particle under oscillating potential barriers, and that time scale is identical to the Büttiker-Landauer traversal time.
The authors analyse tunnelling through time-modulated barriers. The numerical results for the transmitted sideband intensities, obtained with wave-packet simulations, are in quantitative agreement with the analytical results of Buttiker and Landauer (1982, 1985). For opaque barriers the asymmetry of the sideband intensities can be characterized with a single frequency whose inverse Buttiker and Landauer interpret as the traversal time. This simple behaviour does not hold in general, and the authors discuss some of the difficulties one meets in an attempt to generalise the one-parameter description to arbitrary barriers. Furthermore, they examine analytically the transmitted wavefunction in the adiabatic limit, and show that the related density and current involve both the dwell time and the Buttiker time, defined in the context of a Larmor clock. This suggests the possibility that experiments probing different aspects of the transmitted wavefunction may yield different, but complementary, information about the tunnelling process.
We investigate, in one spatial dimension, the quantum mechanical tunnelling of a diatomic, homogeneous molecule with a single bound state incident upon an external barrier. Both time-independent and time-dependent tunnelling are investigated, using analytical and numerical methods. In the time-independent case, we first derive a formal solution for the molecule's wave function. Then, using the method of variable reflection and transmission amplitudes, we find that the probabilities of reflection and transmission in the bound state decrease with decreased binding strength, while the probabilities of refection and transmission in an unbound state increase with decreased binding energy. In the time-dependent case, we consider a molecule with discrete unbound states. The molecular wave function is modeled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration. It is found that, in addition to reflecting and transmitting, the molecule may also straddle the potential barrier in an unbound state.
How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the non-existence of an energy–time commutator analogous to the position–momentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the time-dependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a time-of-flight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.
The tunneling time of particle through given barrier is commonly defined in terms of "internal clocks" which effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Buttiker and Landauer (Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a "clock independent" intrinsic time scale (the resonance life time) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.