Survival and Nonescape Probabilities for Resonant and Nonresonant Decay

TL;DR

This paper analyzes the decay dynamics of a particle in a finite potential, comparing survival and nonescape probabilities, revealing exponential and power-law decay behaviors, and identifying a universal transition to the ground state.

Contribution

It provides an exact solution to the time-dependent Schrödinger equation for finite-range potentials, distinguishing resonant and nonresonant decay behaviors and their long-term dynamics.

Findings

Exponential decay occurs shortly after initial time for resonant states.

Long-time decay follows a power law with different exponents for S(t) and P(t).

Universal transition to the ground state indicates loss of memory in decay processes.

Abstract

In this paper we study the time evolution of the decay process for a particle confined initially in a finite region of space, extending our analysis given recently (Phys. Rev. Lett. 74, 337 (1995)). For this purpose, we solve exactly the time-dependent Schroedinger equation for a finite-range potential. We calculate and compare two quantities: (i) the survival probability S(t), i.e., the probability that the particle is in the initial state after a time t; and (ii) the nonescape probability P(t), i.e., the probability that the particle remains confined inside the potential region after a time t. We analyze in detail the resonant and nonresonant decay. In the former case, after a very short time, S(t) and P(t) decay exponentially, but for very long times they decay as a power law, albeit with different exponents. For the nonresonant case we obtain that both quantities differ initially. However, independently of the resonant and nonresonant character of the initial state we always find a transition to the ground state of the system which indicates a process of ``loss of memory'' in the decay.