Number of quantal resonances

Abstract

Based on extraction of resonances from quantal time delay, a theorem relating quantal time-delay and the number of resonances below a certain energy is proved here. Several illustrations from quantum mechanics, neutron reflectometry and hadron resonances are presented.PACS numbers: 03.65.NkPhysics of weakly-bound systems and resonances has been of great interest for the important role it plays in nuclear and particle physics. In particular, deriving reliable information about unstable and short-lived states leads to a deeper understanding in the theory of elementary particles in case of hadron resonances [1] and hypernuclei [2], compound nuclear resonances [3] in (n, γ) and (p, γ) reactions and in development of new models and theories in the active field of radioactive nuclei [4]. The research on halo nuclei is also intimately connected with the search for superheavy nuclei [5]. The structure of these elusive states is usually in the form of two-body, three-body or many-body resonances [6]. It would be interesting if we could infer from the phase-shift data for π-π, or, η-d, or, α-α scattering whether respectively a ρ-meson around E = 770M eV is formed, or, a hyper-nucleus, or, a resonant α-cluster [7] is formed. Clearly, we would like to be sure of our analysis by studying simple, non-trivial illustrations from quantum mechanics. In this paper, we present various illustrations to elucidate how resonant structures can be reliably extracted and understood.The concept of central importance for our considerations is that of timedelay. Time-delay, T is defined as the difference between the density of states with and without interaction. This reduces to the following relation

Resonances and adiabatic invariance in classical and quantum scattering theory

“…This is a very important result as it immediately implies that the integral provides quantization condition for resonances. This explains all the recent results [9,10,11,12] where resonances in non-relativistic quantum mechanical problems , in neutron reflectometry, and in particle physics are reproduced by studying time-delay as a function of energy. Thus, we have shown that quantization (1) is connected to adiabatic invariance, and the correspondence with the phase space volume is established.…”

“…We now focus on an individual quantum resonance and the phase space scenario associated with it. Let us observe that the resonances have more and more width as they occur at higher and higher energies, thus having lesser and lesser lifetimes [9]. Understanding that the energy-dependent mean lifetime is found from time-delay, this situation is the quantum analogue of the adiabatic invariance of product of energy and time-period of an adiabatically perturbed simple pendulum.…”

“…Recently, it was found that the energy-integral of time-delay, τ gives the number of resonances, n R [9]:…”

“…The root of this universality is the adiabatic invariance that transcends classical or quantal description. Further, various resonances of a system are found by the quantization condition (1) which is based on the adiabatic invariance of the integral in (9) and its clear classical interpretation. These ideas have made it possible for us to find a relation between classical exponent γ (deterministic) corresponding to an unstable fixed point, classical friction γ f r (statistical), and half-width of resonance Γ/2 (statistical).…”

Zeptosecond dynamics in atoms: fact or fiction?

“…2. The autoionization process does not occur instantly as the electron has to move from the interaction regime to an interaction free regime, which introduces a time delay because of the difference between the density of states of the two regions [27]. The energy-integral of time delay is an adiabatic invariant in quantum scattering theory and it provides a quantization condition for resonances [28].…”