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CRC Press eBooks

Decay of Metastable States

Hiroki Nakamura, Gennady Mil'nikov

Published Jul 18, 2013• 0 citations
Publisher

Abstract

In the case of decay rate problem, the potential barrier separates the potentialminimum from the continuum where the system is energetically allowed to go to infinity [see Figure 1.1(b)]. The general formulation developed in Chapter 6 for tunneling splitting can be applied to the present problem with some modifications [44]. The instanton trajectoryq0(τ ) starts in the infinitely remote past at τ = −∞ frompotentialminimum and returns back to the minimum in the infinitely remote future at τ = ∞. This periodic orbit always goes through a turning point on the potential contour V (q) = 0, which can be assumed to occur at τ = 0.

DOI
10.1201/b14673-12

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In the case of decay rate problem, the potential barrier separates the potentialminimum from the continuum where the system is energetically allowed to go to infinity [see Figure 1.1(b)]. The general formulation developed in Chapter 6 for tunneling splitting can be applied to the present problem with some modifications [44]. The instanton trajectoryq0(τ ) starts in the infinitely remote past at τ = −∞ frompotentialminimum and returns back to the minimum in the infinitely remote future at τ = ∞. This periodic orbit always goes through a turning point on the potential contour V (q) = 0, which can be assumed to occur at τ = 0.
Abstract In Nature various metastable states appear, i.e., states which, even though they behave like stationary states for a time duration longer than their characteristic periods of quantum motion, after much longer times make transitions to states in the continuum, i.e., decay. It suffices to think of all the excited states of atoms and molecules, or unstable nuclei, to realize the ubiquitous nature of the problem. Many examples of these processes have already been treated in the context of perturbation theory in Section 9.4; an analysis of a particle in external potentials has been made in the semi-classical approximation in Chapter 11.
In the study of quantum-mechanical tunneling processes, numerous approaches have been developed to determine the decay rate of states initially confined within a metastable potential region. Virtually all analytical treatments, however, fall into one of two superficially unrelated conceptual frameworks: the resonant-state approach and the instanton method. Whereas the concept of resonant states and their associated decay widths is grounded in physical reasoning by capturing the regime of uniform probability decay, the instanton method lacks a comparably clear physical interpretation. We demonstrate the equivalence of the two approaches, revealing that the contour-deformation prescription in the functional integral put forward by Callan and Coleman directly corresponds to the outgoing Gamow--Siegert boundary conditions defining resonant states.
Tunneling is a supreme quantum effect. Every introductory text [1] on quantum mechanics gives the paradigm example of a particle tunneling through a one-dimensional potential barrier despite having a total energy less than the barrier height. Indeed, the reader typically works through a number of exercises, all involving one-dimensional potential barriers of one form or another modeling several key physical phenomena ranging from atom transfer reactions to the decay of α-particles [2]. However, one seldom encounters coupled multidimenisonal tunneling in such texts since an analytical solution of the Schro¨dinger equation in such cases is not possible. Interestingly, the richness and complexity of the tunneling phenomenon manifest themselves in full glory in the case of multidimensional systems [3]. Thus, for instance, the usual one-dimensional expectation of increasing tunneling splittings as one approaches the barrier top from below is not necessarily true as soon as one couples another bound degree of freedom to the tunneling coordinate. In the context of molecular reaction dynamics, multidimensional tunneling can result in strong mode-specificity and fluctuations in the reaction rates [4]. In fact, a proper description of tunneling of electrons and hydrogen atoms is absolutely essential [5,6] even in molecular systems as large as enzymes and proteins. Although one usually assumes tunneling effects to be significant in molecules involving light atom transfers, it is worth pointing out that neglecting the tunneling of even a heavy atom like carbon is the difference between a reaction occurring or not occurring. In particular, one can underestimate rates by nearly hundred orders of magnitude [7]. Interestingly, and perhaps paradoxically, several penetrating insights into the nature and mechanism of multidimensional barrier tunneling have been obtained from a phase-space perspective [8,9]. The contributions by Creagh, Shudo and Ikeda, and Takahashi in the present volume provide a detailed account of the latest advances in the phasespace-based understanding of multidimensional barrier tunneling.
Abstract Perturbative expansion can be generated by calculating Euclidean functional integrals by the steepest descent method always looking, in the absence of external sources, for saddle points in the form of constant solutions to the classical field equations. However, classical field equations may have non-constant solutions. In Euclidean stable field theories, non-constant solutions have always a larger action than minimal constant solutions, because the gradient term gives an additional positive contribution. The non-constant solutions whose action is finite, are called instanton solutions and are the saddle points relevant for a calculation, by the steepest descent method, of barrier penetration effects. This chapter is devoted to simple examples of non-relativistic quantum mechanics (QM), where instanton calculus is an alternative to the semi-classical Wentzel–Kramers–Brillouin (WKB) method. The role of instantons in some metastable systems in QM is explained. In particular, instantons determine the decay rate of metastable states in the semi-classical limit initially confined in a relative minimum of a potential and decaying through barrier penetration. The contributions of instantons at leading order for the quartic anharmonic oscillator with negative coupling are calculate explicitly. The method is generalized to a large class of analytic potentials, and explicit expressions, at leading order, for one-dimensional systems are obtained.