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:2108.07838 · Physics Letters B

Multichannel decay law

Francesco Giacosa

Published May 26, 2022• 12 citations
arXivHTMLPDF

Abstract

It is well known, both theoretically and experimentally, that the survival probability for an unstable quantum state, formed at t = 0 , is not a simple exponential function, even if the latter is a good approximation for intermediate times. Typically, unstable quantum states/particles can decay in more than a single decay channel. In this work, the general expression for the probability that an unstable state decays into a certain i -th channel between the initial time t = 0 and an arbitrary t > 0 is provided, both for nonrelativistic quantum states and for relativistic particles . These partial decay probabilities are also not exponential and their ratio turns out to be not a simple constant, as it would be in the exponential limit. Quite remarkably, these deviations may last relatively long, thus making them potentially interesting in applications. Thus, multichannel decays represent a promising and yet unexplored framework to search for deviations from the exponential decay law in quantum mechanical systems, such as quantum tunneling, and in the context of particle decays .

DOI
10.1016/j.physletb.2022.137200

Related papers

Multichannel Decay: Alternative Derivation of the i-th Channel Decay Probability

Francesco Giacosa

In the study of decays, it is quite common that an unstable quantum state/particle has multiple distinct decay channels.In this case, besides the survival probability p(t), also the probability wi(t) that a decay occurs between (0, t) in the i-th channel is a relevant object.The general form of the function wi(t) was recently presented in PLB 831, 137200 (2022).Here, we provide a novel and detailed "joint" derivation of both p(t) and wi(t).As it is well known, p(t) is not an exponential function; similarly, wi(t) is not one either.In particular, the ratio wi/wj (for i = j) is not a simple constant as it would be in the exponential limit.The functions wi(t) and their mutual ratios may therefore represent a novel tool to study the non-exponential nature of the decay law.

arXiv:2207.13977•Acta Physica Polonica A•Sep 1, 2022•1 citation

Capturing nonexponential dynamics in the presence of two decay channels

Francesco Giacosa, Przemysław Kościk, Tomasz Sowiński

The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the nonexponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through specific quantum mechanical examples which can be accurately solved, we study the general properties of a two-channel decay that apply for various unstable quantum states (including unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit), but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential-decay law.

arXiv:1912.06394•Physical Review A•Aug 11, 2020•14 citations

Nonclassical face of quantum decay processes

K. Urbanowski

A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).

Journal of Physics Conference Series•Aug 1, 2017•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

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

Properties of decaying relativistic particles: "GSI anomaly" like effects

K. Urbanowski

We study the survival probability of moving relativistic unstable particles and we found the integral representation of the survival amplitude for such particles moving with constant momentum ~ 6 0. Using this representation of the amplitude we found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t/γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities tend to zero as t → ∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the properly computed decay curves of the moving particles have analogous fluctuating form as the decay curves obtained in the GSI experiment.

•Jan 1, 2015•1 citation
It is well known, both theoretically and experimentally, that the survival probability for an unstable quantum state, formed at t = 0 , is not a simple exponential function, even if the latter is a good approximation for intermediate times. Typically, unstable quantum states/particles can decay in more than a single decay channel. In this work, the general expression for the probability that an unstable state decays into a certain i -th channel between the initial time t = 0 and an arbitrary t > 0 is provided, both for nonrelativistic quantum states and for relativistic particles . These partial decay probabilities are also not exponential and their ratio turns out to be not a simple constant, as it would be in the exponential limit. Quite remarkably, these deviations may last relatively long, thus making them potentially interesting in applications. Thus, multichannel decays represent a promising and yet unexplored framework to search for deviations from the exponential decay law in quantum mechanical systems, such as quantum tunneling, and in the context of particle decays .
In the study of decays, it is quite common that an unstable quantum state/particle has multiple distinct decay channels.In this case, besides the survival probability p(t), also the probability wi(t) that a decay occurs between (0, t) in the i-th channel is a relevant object.The general form of the function wi(t) was recently presented in PLB 831, 137200 (2022).Here, we provide a novel and detailed "joint" derivation of both p(t) and wi(t).As it is well known, p(t) is not an exponential function; similarly, wi(t) is not one either.In particular, the ratio wi/wj (for i = j) is not a simple constant as it would be in the exponential limit.The functions wi(t) and their mutual ratios may therefore represent a novel tool to study the non-exponential nature of the decay law.
The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the nonexponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through specific quantum mechanical examples which can be accurately solved, we study the general properties of a two-channel decay that apply for various unstable quantum states (including unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit), but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential-decay law.
A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).
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.
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 survival probability of moving relativistic unstable particles and we found the integral representation of the survival amplitude for such particles moving with constant momentum ~ 6 0. Using this representation of the amplitude we found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t/γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities tend to zero as t → ∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the properly computed decay curves of the moving particles have analogous fluctuating form as the decay curves obtained in the GSI experiment.