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The seventh international conference on hadron spectroscopy

Finite-size corrections to Fermi's golden rule

Kenzo Ishikawa, Yutaka Tobita

Published Apr 7, 2014• 0 citations
Publisher

Abstract

The transition process in quantum mechanics has been studied with the probability ΓT proportional to the time interval T between the initial and final states, where Γ is the rate.Recently it was found that the constant term P (d) is added as P = ΓT + P (d) .P (d) has been ignored for a macroscopic T, but that is not ignoable in various systems.We find P (d) in such processes that Γ ≈ 0 and show that it has wave-like properties and leads unusual quantum phenomena.P (d) is large in magnitude and in spatial size for light particles and has universal properties in relativisticaly invariant systems.Implications to processes of neutrinos in beta decays and gamma rays in various radiative transitions are presented.

DOI
10.22323/1.205.0149

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The transition process in quantum mechanics has been studied with the probability ΓT proportional to the time interval T between the initial and final states, where Γ is the rate.Recently it was found that the constant term P (d) is added as P = ΓT + P (d) .P (d) has been ignored for a macroscopic T, but that is not ignoable in various systems.We find P (d) in such processes that Γ ≈ 0 and show that it has wave-like properties and leads unusual quantum phenomena.P (d) is large in magnitude and in spatial size for light particles and has universal properties in relativisticaly invariant systems.Implications to processes of neutrinos in beta decays and gamma rays in various radiative transitions are presented.
A quantum mechanical wave of a finite size moves like a classical particle and shows a unique decay probability. Because the wave function evolves according to the Schrodinger equation, it preserves the total energy but not the kinetic energy in the intermediate-time region of a decay process where those of the parent and daughters overlap. The decay rate computed with Fermi's golden rule requires corrections that vary with the distance between the initial and final states, and the energy distribution of the daughter is distorted from that of plane waves. The corrections have universal properties in relativistically invariant systems and reveal macroscopic quantum phenomena for light particles. The implications for precision experiments in beta decays and various radiative transitions are presented.
A quantum mechanical wave of a finite size moves like a classical particle and shows a unique decay probability. Because the wave function evolves according to the Schrödinger equation, it preserves the total energy but not the kinetic energy in the intermediate-time region of a decay process where those of the parent and daughters overlap. The decay rate computed with Fermi's golden rule requires corrections that vary with the distance between the initial and final states, and the energy distribution of the daughter is distorted from that of plane waves. The corrections have universal properties in relativistically invariant systems and reveal macroscopic quantum phenomena for light particles. The implications for precision experiments in beta decays and various radiative transitions are presented.
Many-body states described by a Schrödinger equation include states of overlapping waves of non-vanishing interaction energies. These peculiar states formed in many-body transitions remain in asymptotic regions, and lead a new component to the transition probability. The probability is computed rigorously following the von Neumann's fundamental principle of quantum mechanics with an S-matrix that is defined with normalized functions, instead of plane waves. That includes the intriguing correction term to the Fermi's golden rule, in which a visible energy is smaller than the initial energy, and reveals macroscopic quantum phenomena for light particles. Processes in Quantum Electrodynamics are analyzed and the sizable corrections are found in the dilute systems. The results suggest that these states play important roles in natural phenomena, and the verification in laboratory would be possible with recent advanced technology.
Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.
Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.
Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.