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Metastable states

Kenichi Konishi, Giampiero Paffuti

Published Mar 5, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780199560264.003.0013

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In many practical cases a stationary system is perturbed in some way, inducing it to undergo transitions between the otherwise stationary states. Perhaps the best example concerns the radiative transitions of atomic electrons in which the disturbance of an external electromagnetic field causes the electrons to jump between energy levels, emitting or absorbing photons. Equation is a statement of the classical law of conservation of energy. The stimulated emission and absorption of radiation both have close classical analogues. If a simple harmonic oscillator is driven by a sinusoidal electromagnetic field it can gain or lose energy depending on phase angles. The existence of spontaneous emission implies that all excited states of electrical systems are unstable and will eventually decay to the ground state by photon emission. Quantum systems thus comply with the fundamental principle that physical systems tend to seek out their state of lowest energy and maximum entropy.
Introduction We now move to considering the dynamics of a system of nuclei and electrons. Of course, both electrons and nuclei are subject to the laws of quantum mechanics, but since nuclei are 2000 to 200,000 times heavier than electrons, we expect that classical mechanics will be a much better approximation for the motion of nuclei than for the motion of electrons. This means that we expect a level of approximation to be valid, where some of the degrees of freedom (d.o.f.) of a system behave essentially classically and others behave essentially quantum-mechanically. The system then is of a mixed quantum/classical nature. Most often the quantum subsystem consists of system of electrons in a dynamical field of classical nuclei, but the quantum subsystem may also be a selection of generalized nuclear coordinates (e.g., corresponding to high-frequency vibrations) while other generalized coordinates are supposed to behave classically, or describe the motion of a proton in a classical environment. So, in this chapter we consider the dynamics of a quantum system in a non-stationary potential. In Section 5.2 we consider the time-dependent potential as externally given, without taking notice of the fact that the sources of the time-dependent potential are moving nuclei, which are quantum particles themselves, feeling the interaction with the quantum d.o.f. Thus we consider the time evolution of the quantum system, which now involves mixing-in of excited states, but we completely ignore the back reaction of the quantum system onto the d.o.f. that cause the time-dependent potential, i.e., the moving nuclei.
Abstract Avoided crossings of level pairs with opposite slopes can form potential-energy minima for the external degree of freedom of quantum particles, giving rise to metastable states on the avoided crossings (MSACs). Nonadiabatic decay of MSACs is studied by solving the two-component Schrödinger equation in diabatic and adiabatic representations. Non-perturbative lifetime values are found by evaluating wave function flux and scattering phases of time-independent solutions, as well as wave-function decay of time-dependent solutions. The values from these methods generally agree well, validating the utilized approaches. As the adiabaticity parameter, V, of the system is increased by about a factor of ten across the mixed diabatic/adiabatic regime, the MSAC character transitions from marginally to highly stable, with the lifetimes increasing by about ten orders of magnitude. The dependence of MSAC lifetime on the vibrational quantum number, ν, is discussed for several regimes of V. Time-dependent perturbation theory yields lifetimes that deviate by ≲30% from non-perturbative results, over the range of V and ν studied, while a semi-classical model based on Landau–Zener tunneling is up to a factor of twenty off. The results are relevant to numerous atomic and molecular systems with metastable states on intersecting, coupled potential energy curves.