Theory of Quantum Resonances II: The Shape Resonance Model

Abstract

The shape resonance model was developed by Gamow [Ga] and by Gurney and Condon [GC] to describe the decay of an unstable atomic nucleus by alpha-particle emission. The idea is very simple. The atomic nucleus is modeled by a potential barrier of finite width which traps the alpha particle. A typical situation is shown in Figure 16.1. According to quantum theory, the wave function for the alpha particle, initially localized in the potential well between the barriers, will oscillate between the barriers. However, because the barriers have finite thickness, as measured by the Agmon metric, the wave function penetrates the barriers, and hence the particle has a nonzero probability of escaping to infinity. In fact, the probability that the particle will escape to infinity in infinite time is 1. To say this another way, there are typically no bound states for this model, and consequently a decay condition such as (16.1) holds: The wave function will eventually leave every bounded region. The classical limit of this model is clear: The barriers are infinitely high, so the distance in the Agmon metric across the barrier is infinite. This forces the wave function to vanish in the classically forbidden region, and the alpha particle is in a bound state. As we will show by a rescaling of the Schrödinger operator, this is equivalent to taking Planck’s constant to zero. The semiclassical regime, therefore, is described by very large potential barriers relative to the energy of the alpha particle. We describe the alpha particle as a quantum resonance of a Hamiltonian H(λ) = -△ + λ² V, where V has the “shape” of a confining potential barrier of finite thickness determined by λ. We will work in the large λ regime.