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Physical Review A

Tunneling of a diatomic molecule incident upon a potential barrier

Glen L. Goodvin, Mark R. A. Shegelski

Published Mar 25, 2005• 27 citations
Publisher

Abstract

We study a diatomic homonuclear molecule incident upon a potential barrier. We allow transitions between binding energy states of the molecule during the process of tunneling and consider both single and double barriers. By applying an elegant method to solve this system we show that transmission resonances do occur in both cases and that expressions for the locations of the resonances can be found. Moreover, we find that the addition of binding energy levels tends to decrease the probability of tunneling. We also show that by relaxing rigor and treating the length of the molecule as though it is fixed we can obtain exact analytical expressions for the transmission probability that capture important features of the full solution.

DOI
10.1103/physreva.71.032719

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Tunneling of a diatomic molecule

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We summarize our study of a diatomic homonuclear molecule incident upon a potential barrier in one dimension [G.L. Goodvin and M.R.A. Shegelski. Phys. Rev. A, 71, 032719 (2005).]. We study various initial states for the molecule and allow transitions between vibrational states of the molecule during the process of tunneling. By applying an elegant method to solve this problem, we show that transmission resonances occur in several cases and that adding more binding energy levels for the molecule can tend to decrease the probability of tunneling. We also discuss the extension of our results to more realistic three-dimensional and experimental systems.PACS Nos.: 34.20.Cf, 03.65.Xp, 34.20.Gj, 34.50.Pi

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We study the effects of multiple bound states on the probability of tunneling of a homonuclear diatomic molecule incident upon a potential barrier in one dimension. We obtain additional bound states by several means: by varying individual features of the binding potential as well as increasing the mass of the two atoms. We find that the tunneling behavior of a molecule with a large number of bound states $(\ensuremath{\sim}30)$ is drastically different from that for a molecule with a small number of states $(\ensuremath{\lesssim}5)$. Further, we find that the effect of multiple bound states on the proabability of tunneling is largely dependent upon the separation between the bound-state energies. It is also found that intermediary transitions between many bound states results in significantly larger reductions in the probability of tunneling. Finally, we investigate molecules of different masses and observe common qualitative features. We focus on molecules incident upon a $\ensuremath{\delta}$ barrier but we also consider Gaussian barriers.

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We study in three dimensions a diatomic homo-nuclear molecule with many bound states incident upon a potential barrier. We consider different initial states for the molecule and take into account transitions between rotational states of the molecule during the process in which the entire molecule tunnels past or is reflected by the external potential barrier. We show the manner in which the transmission resonances are affected by having many bound states. We find that the transmission probability profile in three dimensions is markedly different from that for two bound states in three dimensions, and for many bound states in one dimension. We find important new results. One result is that resonant or near resonant tunnelling and resonant or near resonant reflection can both occur within a small energy range of the molecule. We also find that tunnelling with transmission probability close to unity can occur over a wide energy range.

Journal of Physics B Atomic Molecular and Optical Physics•Feb 4, 2013•4 citations
We study a diatomic homonuclear molecule incident upon a potential barrier. We allow transitions between binding energy states of the molecule during the process of tunneling and consider both single and double barriers. By applying an elegant method to solve this system we show that transmission resonances do occur in both cases and that expressions for the locations of the resonances can be found. Moreover, we find that the addition of binding energy levels tends to decrease the probability of tunneling. We also show that by relaxing rigor and treating the length of the molecule as though it is fixed we can obtain exact analytical expressions for the transmission probability that capture important features of the full solution.
We summarize our study of a diatomic homonuclear molecule incident upon a potential barrier in one dimension [G.L. Goodvin and M.R.A. Shegelski. Phys. Rev. A, 71, 032719 (2005).]. We study various initial states for the molecule and allow transitions between vibrational states of the molecule during the process of tunneling. By applying an elegant method to solve this problem, we show that transmission resonances occur in several cases and that adding more binding energy levels for the molecule can tend to decrease the probability of tunneling. We also discuss the extension of our results to more realistic three-dimensional and experimental systems.PACS Nos.: 34.20.Cf, 03.65.Xp, 34.20.Gj, 34.50.Pi
We study a diatomic homonuclear molecule incident upon a potential barrier in three dimensions. We study various initial states for the molecule and allow transitions between rotational states of the molecule during the process of tunneling. By applying an elegant method to solve this problem we show that transmission resonances occur in several cases and that adding more binding energy levels for the molecule can tend to decrease the probability of tunneling. We also find that the probability of transmission can be enhanced by the addition of a second potential barrier. These results are in qualitative agreement with those found for the tunneling of a molecule in a one-dimensional model studied previously.
We study the reflection and transmission of a homonuclear diatomic molecule incident upon a potential barrier in one dimension. The effect of discrete and continuous unbound molecular states is investigated. We use the method of variable reflection and transmission amplitudes for discrete unbound states and we extend the method to include continuous unbound states. We take into account transitions between the bound and unbound states in the process of tunneling. For the molecule incident in a bound state, we calculate the probabilities of reflection and transmission in bound states as well as in unbound states. We focus on the molecule incident upon a $\ensuremath{\delta}$ barrier but we also investigate rectangular and Gaussian barriers. We show that transmission resonances are appreciably reduced by the inclusion of unbound states due to the lack of resonant structure in the probabilities of reflection and transmission in unbound states. We also find that much of the behavior of the molecule in the process of tunneling is primarily due to the bound states.
We study the effects of multiple bound states on the probability of tunneling of a homonuclear diatomic molecule incident upon a potential barrier in one dimension. We obtain additional bound states by several means: by varying individual features of the binding potential as well as increasing the mass of the two atoms. We find that the tunneling behavior of a molecule with a large number of bound states $(\ensuremath{\sim}30)$ is drastically different from that for a molecule with a small number of states $(\ensuremath{\lesssim}5)$. Further, we find that the effect of multiple bound states on the proabability of tunneling is largely dependent upon the separation between the bound-state energies. It is also found that intermediary transitions between many bound states results in significantly larger reductions in the probability of tunneling. Finally, we investigate molecules of different masses and observe common qualitative features. We focus on molecules incident upon a $\ensuremath{\delta}$ barrier but we also consider Gaussian barriers.
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.
We study in three dimensions a diatomic homo-nuclear molecule with many bound states incident upon a potential barrier. We consider different initial states for the molecule and take into account transitions between rotational states of the molecule during the process in which the entire molecule tunnels past or is reflected by the external potential barrier. We show the manner in which the transmission resonances are affected by having many bound states. We find that the transmission probability profile in three dimensions is markedly different from that for two bound states in three dimensions, and for many bound states in one dimension. We find important new results. One result is that resonant or near resonant tunnelling and resonant or near resonant reflection can both occur within a small energy range of the molecule. We also find that tunnelling with transmission probability close to unity can occur over a wide energy range.