Irreversible Evolution of a Wave Packet in The Rigged Hilbert Space Quantum Mechanics

TL;DR

This paper explores the irreversible evolution of wave packets in Rigged Hilbert Space quantum mechanics, focusing on Gamow Vectors of the Reversed Harmonic Oscillator to model decay and irreversibility.

Contribution

It introduces a detailed analysis of wave packet evolution using Rigged Hilbert Space formalism, including background functions and numerical studies, to better understand irreversible quantum dynamics.

Findings

Gamow Vectors effectively model decay processes.

Wave packets exhibit irreversible evolution in RHS.

Background functions help analyze complex initial data.

Abstract

It is well known that a state with complex energy cannot be the eigenstate of a self-adjoint operator, like the Hamiltonian. Resonances, i.e. states with exponentially decaying observables, are not vectors belonging to the conventional Hilbert space. One can describe these resonances in an unusual mathematical formalism, based on the so-called Rigged Hilbert Space (RHS). In the RHS, the states with complex energy are denoted as Gamow Vectors (GV), and they model decay processes. We study GV of the Reversed Harmonic Oscillator (RHO), and we analytically and numerically investigate the unstable evolution of wave packets. We introduce the background function to study initial data not composed only by a summation of GV and we analyse different wave packets belonging to specific function spaces. Our work furnishes support to the idea that irreversible wave propagations can be investigated by means of Rigged Hilbert Space Quantum Mechanics and provides insights for the experimental investigation of irreversible dynamics.