Time-domain resonances and the ultimate fate of a decaying quantum state

Abstract

We derive an analytical expression for the propagation of a quantum particle at asymptotic long distances and times from the potential where the particle was initially confined. We obtain that the particle is described by an evolving resonance in time domain that possesses a peaked shape characterized by the resonance parameters of the dominant resonance coefficient of the decaying wave function. The above situation corresponds to an unexplored postexponential regime that represents the ultimate fate of a decaying particle.

Heisenberg uncertainty relations for the non-Hermitian resonance-state solutions to the Schrödinger equation

“…The present work shows that the Heisenberg uncertainty relations may hold beyond the standard Hermitian framework of quantum mechanics. This might be of particular interest for those pursuing a line of inquiry that explores the possibility of extending the standard formalism of quantum mechanics to incorporate in a fundamental fashion a non-Hermitian treatment of the Hamiltonian to the system, as suggested by studies on tunneling decay [34,35].…”

Quantum decay law: critical times and the equivalence of approaches

“…Another method for obtaining the survival amplitude is to solve (1) using Green's functions. Finding the Green's function may be a laborious undertaking, however, there exists an elegant approach proposed by Garcia-Calderon (GC) [10] (and followed up in [11,15,16,20,37]) which overcomes this difficulty. The GC approach uses resonant states for calculating the Green's function, the corresponding wave function and the survival amplitude.…”

“…In order to at least partly answer the above questions, it is essential to investigate if the different formalisms agree only globally on the short and large time behaviour or also in details such as the prediction of the critical transition times from the exponential to the power law as well as the exponent in the power law behaviour. With this objective, in the present work, we investigate some of the most popularly known approaches for the calculation of survival probabilities, namely, the method of García-Calderón (GC) [10,16,17] and collaborators [11,15,[18][19][20] which uses the Green's functions, the method of W. van Dijk and Y. Nogami (DN) [13,14] and the Fock-Krylov (FK) method [9] which involves the Fourier transform of an energy density. We show that the seemingly different methods are indeed equivalent and derive analytical expressions for the survival amplitudes as well as their large time behaviour.…”

Underlying non‐Hermitian character of the Born rule in open quantum systems

“…In particular, using some properties of the function , the survival amplitude may be written for the exponential and long times regimes as , or to discuss the ultimate fate of a decaying quantum state .…”