Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1708.02083 · Acta Physica Polonica B

Time Evolution of an Unstable Quantum System

Francesco Giacosa

Published Jan 1, 2017• 8 citations
arXivHTMLPDF

Abstract

After reviewing the description of an unstable state in the framework of Lee Hamiltonians (valid both for Quantum Mechanics (QM) and Quantum Field Theory (QFT)), we consider some theoretical aspects of non-exponential decays: the case of two decay channels, the broadening of the energy spectrum at short times, the effect of an imperfect measurement, the link to QFT, and the decay of an unstable moving particle with definite momentum. All the presented effects were not confirmed in experiments, hence are at the present stage predictions.

DOI
10.5506/aphyspolb.48.1831

Related papers

Non-exponential decay in Quantum Mechanics and Quantum Field Theory

Francesco Giacosa

We describe some salient features as well as some recent developments concerning short-time deviations from the exponential decay law in the context of Quantum Mechanics by using the Lee Hamiltonian approach and Quantum Field Theory by using relativistic Lagrangians. In particular, the case in which two decay channels are present is analyzed: the ratio of decay probability densities, which is a constant equal to the ratio of decay widths in the exponential limit, shows in general sizable fluctuations which persist also at long times.

arXiv:1312.3315•Journal of Physics Conference Series•Oct 10, 2014•2 citations

Toward a QFT treatment of nonexponential decay

Francesco Giacosa

We study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.

arXiv:1711.10727•EPJ Web of Conferences•Jan 1, 2018•0 citations

Toward a QFT treatment of nonexponential decay

Francesco Giacosa

\nWe study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.\n

•Aug 3, 2018•0 citations

Decay Law and Time Dilatation

Francesco Giacosa

We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum $p$ and mass $M$ is $\tildeΓ_{p}=ΓM/\sqrt{p^{2}+M^{2}}$ with $Γ$ being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT this equation is \textit{not} correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum $p$ (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula and we show that the deviation from $\tildeΓ_{p}$ has a maximum for $p/M=\sqrt{2/3},$ but is typically \textit{very} small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that $\tilde{Γ}_{p}$ is typically a very good approximation. The study of a velocity wave-packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript.\textbf{ }In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.

arXiv:1512.00232•Acta Physica Polonica B•Jan 1, 2016•17 citations

The Lee model: a tool to study decays

Francesco Giacosa

Abstract We describe- in a didactical and detailed way- the so-called Lee model (which shares similarities with the Jaynes-Cummings and Friedrichs models) as a tool to study unstable quantum states/particles. This Lee model is based on Quantum Mechanics (QM) but possesses some of the features of Quantum Field Theory (QFT). The decay process can be studied in great detail and typical QFT quantities such as propagator, one-loop resummation, and Feynman rules can be introduced. Deviations from the exponential decay law as well as the Quantum Zeno effects can be studied within this framework. The survival probability amplitude as a Fourier transform of the energy distribution, the normalization of the latter, and the Breit-Wigner limit can be obtained in a rigorous mathematical approach.

arXiv:2001.07781•Journal of Physics Conference Series•Aug 1, 2020•5 citations

Influence of the measurement on the decay law: the bang-bang case

Francesco Giacosa, Giuseppe Pagliara

After reviewing the description of an unstable state in the framework of nonrelativistic Quantum Mechanics (QM) and relativistic Quantum Field Theory (QFT), we consider the effect of pulsed, ideal measurements repeated at equal time intervals on the lifetime of an unstable system. In particular, we investigate the case in which the ‘bare’ survival probability is an exact exponential (a very good approximation in both QM and QFT), but the measurement apparatus can detect the decay products only in a certain energy range. We show that the Quantum Zeno Effect can occur in this framework as well.

EPJ Web of Conferences•Jan 1, 2015•2 citations
After reviewing the description of an unstable state in the framework of Lee Hamiltonians (valid both for Quantum Mechanics (QM) and Quantum Field Theory (QFT)), we consider some theoretical aspects of non-exponential decays: the case of two decay channels, the broadening of the energy spectrum at short times, the effect of an imperfect measurement, the link to QFT, and the decay of an unstable moving particle with definite momentum. All the presented effects were not confirmed in experiments, hence are at the present stage predictions.
We describe some salient features as well as some recent developments concerning short-time deviations from the exponential decay law in the context of Quantum Mechanics by using the Lee Hamiltonian approach and Quantum Field Theory by using relativistic Lagrangians. In particular, the case in which two decay channels are present is analyzed: the ratio of decay probability densities, which is a constant equal to the ratio of decay widths in the exponential limit, shows in general sizable fluctuations which persist also at long times.
We study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.
\nWe study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.\n
We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum ppp and mass MMM is Γ~p=ΓM/p2+M2\tildeΓ_{p}=ΓM/\sqrt{p^{2}+M^{2}}Γ~p​=ΓM/p2+M2​ with ΓΓΓ being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT this equation is \textit{not} correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum ppp (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula and we show that the deviation from Γ~p\tildeΓ_{p}Γ~p​ has a maximum for p/M=2/3,p/M=\sqrt{2/3},p/M=2/3​, but is typically \textit{very} small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that Γ~p\tilde{Γ}_{p}Γ~p​ is typically a very good approximation. The study of a velocity wave-packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript.\textbf{ }In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.
Abstract We describe- in a didactical and detailed way- the so-called Lee model (which shares similarities with the Jaynes-Cummings and Friedrichs models) as a tool to study unstable quantum states/particles. This Lee model is based on Quantum Mechanics (QM) but possesses some of the features of Quantum Field Theory (QFT). The decay process can be studied in great detail and typical QFT quantities such as propagator, one-loop resummation, and Feynman rules can be introduced. Deviations from the exponential decay law as well as the Quantum Zeno effects can be studied within this framework. The survival probability amplitude as a Fourier transform of the energy distribution, the normalization of the latter, and the Breit-Wigner limit can be obtained in a rigorous mathematical approach.
After reviewing the description of an unstable state in the framework of nonrelativistic Quantum Mechanics (QM) and relativistic Quantum Field Theory (QFT), we consider the effect of pulsed, ideal measurements repeated at equal time intervals on the lifetime of an unstable system. In particular, we investigate the case in which the ‘bare’ survival probability is an exact exponential (a very good approximation in both QM and QFT), but the measurement apparatus can detect the decay products only in a certain energy range. We show that the Quantum Zeno Effect can occur in this framework as well.