| half-life | |
|---|---|
| Name | Half-life |
| Unit | second (s) |
| Dimension | Time |
half-life
Half-life is the time required for a quantity undergoing decay to decrease to half its initial value; in Quantum Physics it characterizes the exponential decrease of unstable quantum states, radioactive nuclei, and unstable particles. It matters because it links microscopic transition probabilities calculated by Quantum mechanics and Quantum field theory to macroscopic observables used in nuclear physics, particle physics, radiometric dating, and medical applications. Half-life encapsulates information about interaction strengths, decay channels, and environmental effects on quantum systems.
In physics the half-life t_{1/2} of an ensemble of identical unstable systems is defined by N(t_{1/2}) = N(0)/2 for the number N(t) of surviving systems. In radiation and nuclear physics contexts it quantifies the lifetime of radionuclides such as uranium-238, carbon-14, and medically relevant isotopes like technetium-99m. In particle physics it is related to the mean lifetime τ of an unstable particle (for exponential decay t_{1/2} = τ ln 2). Half-life gives an operational meaning to decay rates and transition matrix elements computed in models such as the Fermi's golden rule framework and in full quantum field theory calculations (e.g., within the Standard Model).
For a simple exponential process N(t)=N(0)e^{-λt}, the decay constant λ and half-life are related by t_{1/2} = (ln 2)/λ. The probability density for decay at time t is λ e^{-λt}; the survival probability S(t) equals e^{-λt}. This formalism appears across disciplines: radioactive decay, excited-state emission in atomic physics, and decay of quasiparticles in condensed matter. The decay constant λ can be expressed via transition rates from perturbation theory; in many-body contexts it is connected to spectral densities and complex poles of propagators in Green's functions.
Quantum mechanically, decay is described by the survival amplitude a(t)=⟨ψ|e^{-iHt/ħ}|ψ⟩ for an initial unstable state |ψ⟩ evolving under the Hamiltonian H. The survival probability S(t)=|a(t)|^2 yields nontrivial time dependence derived from the energy distribution (spectral function) ρ(E)=|⟨E|ψ⟩|^2. Exponential decay corresponds to a Breit–Wigner (Lorentzian) energy distribution and to a simple pole of the resolvent in the complex energy plane; this links half-life to the imaginary part of complex energy eigenvalues (resonances) studied in scattering theory and S-matrix theory. Techniques from non-Hermitian quantum mechanics and the theory of resonances (e.g., Gamow vectors) provide formal derivations of lifetimes used in calculations by groups at institutions such as CERN, Lawrence Berkeley National Laboratory, and Los Alamos National Laboratory.
Experimentally accessible decay rates Γ are often quoted where Γ = ħ/τ and τ = mean lifetime; for exponential decay t_{1/2}= (ln 2) τ. In nuclear magnetic resonance and muon decay measurements, lifetimes connect to fundamental interactions: weak interaction parameters are extracted from muon lifetime experiments performed at facilities like Paul Scherrer Institute or Fermilab. In atomic physics, spontaneous emission half-lives are determined by coupling to the electromagnetic vacuum and computed via quantum electrodynamics (QED). Detectors and counting statistics, governed by Poisson distribution and Bayesian inference methods, translate raw counts into decay constants and half-lives.
Half-life underpins radiometric dating methods such as carbon-14 dating used in archaeology and geology, where precise half-life values calibrate chronologies. In nuclear engineering and reactor physics, inventories of fissile isotopes and waste management rely on half-lives of actinides and fission products. In particle physics, resonance widths and lifetimes of hadrons (e.g., Δ(1232), J/ψ) inform strong interaction dynamics and are measured in experiments at Large Hadron Collider detectors like ATLAS and CMS. Medical applications include radionuclide therapy and diagnostic imaging with isotopes selected for appropriate half-lives to balance dose and diagnostic yield.
Deviations from strict exponential decay occur at very short and very long times due to the full energy spectrum and unitarity of quantum evolution. The short-time quadratic behavior leads to the quantum Zeno effect—frequent measurements can inhibit decay—first predicted by Baute, Misra and Sudarshan and observed in systems such as trapped ions and cold atoms at institutions like MIT and Max Planck Institute for Quantum Optics. Conversely, the quantum anti-Zeno effect can enhance decay under different measurement regimes. These phenomena show that half-life is not an immutable scalar but can be modified by measurement, environment (decoherence), and engineered reservoirs studied in open quantum systems and quantum optics.
Half-life measurement techniques include direct counting of decays with scintillation counters and semiconductor detectors (e.g., Geiger–Müller tube, HPGe detector), decay-curve fitting, accelerator-based lifetime measurements using time-of-flight and vertex detectors, and indirect spectroscopic methods extracting resonance widths from cross sections. Calibration against standards from agencies like the International Atomic Energy Agency and statistical treatment using maximum-likelihood estimation and confidence intervals are standard practice. High-precision determinations—such as the muon lifetime measured at MuLan (an experiment at Paul Scherrer Institute and PSI) or isotope half-lives refined by NIST—provide inputs to fundamental constants and tests of theoretical models.
Category:Quantum mechanics Category:Nuclear physics Category:Particle physics