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Finite-size corrections to Fermi's golden rule: I. Decay rates

Kenzo Ishikawa, Yutaka Tobita

Published Jul 1, 2013• 15 citations
OpenAlex

Abstract

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.

Related papers

Finite-size corrections to Fermi's golden rule: I. Decay rates

K. Ishikawa, Y. Tobita

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.

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Finite-size corrections to Fermi's golden rule

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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.

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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.

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Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.

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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.
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.
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.
We calculate the scattering rate of a quantum particle, based on Fermi’s golden rule for a quantum particle as a distribution of matter, where the density of states is a function of the particle Lagrangian, not of the Hamiltonian, as in the conventional theory based on the Schrödinger equation. Thus, the density of states does not refer to the energies as in the conventional form, but to the Lagrangian functions, which, besides the energies, include the momentum-velocity products, that are also constants of motion for the initial and the final states. We obtain the relativistic dependence on the momentum, and the two scattering rates for the two possible cases, with the spin conservation, or the spin inversion.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
An elementary account is given of the absorption and emission of radiation by molecules employing a particular formulation of Schrödinger theory which facilitates the treatment of radiative line widths and level shifts. The familiar Fermi golden rule for transition rates, the Rabi Bloch-Siegert absorption profile, and the Weisskopf-Wigner approximation for spontaneous decay are obtained under appropriate conditions in the context of a common perspective. Comparisons are made with other recently devised perturbative and nonperturbative formulations of electromagnetic transition phenomena which correctly include radiative line widths and level shifts, and the importance of proper treatment of the normalization and phase of the Schrödinger wave function in this connection is emphasized.