Continuous crossover from quantum-limit-bounded evolution to exponential delay

Exponential decay is a signature property of a vast class of dynamic systems in atomic, molecular, nuclear, and particle physics. From the perspective of quantum theory, however, exponential decay is a nontrivial behavior that is not obedient to the fundamental bounds that quantum mechanics sets for the evolution of dynamic systems. Here we revisit the question of how the existence and the ubiquity of exponentially decaying systems can be reconciled with and better understood from a perspective of the constitutive principles of quantum mechanics. We show that, contrary to a widespread view, the Mandelstam-Tamm idea of the variance of the evolution Hamiltonian, ${(\mathrm{\ensuremath{\Delta}}E)}^{2}$, as a descriptor of quantum dynamics is fully applicable to exponentially decaying systems. We demonstrate that, with a suitable modification of a physical model of an exponentially decaying system, correcting for an unphysical, infinitely fast onset of quantum evolution, the Mandelstam-Tamm treatment provides a powerful resource for the analysis of exponentially decaying systems, revealing a continuous crossover from the Mandelstam-Tamm-bounded dynamics, unfolding on the short timescale, to exponentially decaying dynamics, which may set in, given a suitable physical setting, at later stages of quantum evolution. In a formalism that we develop, the divergence of $\mathrm{\ensuremath{\Delta}}E$ is removed, by allowing for a nonzero evolution onset time, leading to quantum dynamics that is in full obedience to the constitutive principles of quantum theory, including the Mandelstam-Tamm bound, the Mandelstam-Tamm energy-time uncertainty, and the Krylov-Fock theorem.