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

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.

arXiv comment16 pages, 19 figures, revised version

References in corpus (10)

• Combining complex networks and data mining: why and how
• A Pedestrian Introduction to Gamow Vectors
• Time Asymmetric Quantum Physics
• Quantum Mechanics of Damped Systems II. Damping and Parabolic Potential Barrier
• Quantum mechanics of damped systems
• Schroedinger-like PageRank equation and localization in the WWW
• Nonlinear Gamow vectors, shock waves and irreversibility in optically nonlocal media
• Renormalization in Winter Model
• Metriplectic Algebra for Dissipative Fluids in Lagrangian Formulation
• Optical realization of the dissipative quantum oscillator

Cited by in corpus (10)

• Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons
• Optical Spatial Shock Waves in Nonlocal Nonlinear Media
• Spherical harmonics and rigged Hilbert spaces
• Anisotropic Optical Shock Waves in Isotropic Media with Giant Nonlocal Nonlinearity
• Squeezing in a nonlocal photon fluid
• Unstable mode and the Unruh-DeWitt detector
• Gamow vectors formalism applied to the Loschmidt echo
• Swanson Hamiltonian: non-PT-symmetry phase
• Green's Function Formalism for Impurity-Induced Resonances in Sub-barrier Proton-Nucleus Scattering
• A Swanson-like Hamiltonian and the inverted harmonic oscillator