Resonances I: An ensemble of neutron decays, observed from time t=0, will exhibit the characteristic radioactive decay law:
N(t)=N(0)e−t/τ,(1.139)
where τ=878.4±0.5 s (Review of Particle Properties, 2022) is the mean lifetime of the neutron (convince yourself that this should be the case). Also, if an ensemble of neutrons were each weighed very precisely, it would be found that the mass distribution has a (small) width. The full width at half maximum (FWHM) of this distribution is typically denoted by Γ. The mean lifetime is inversely related to this width: τ=1/Γ.
(a) What is the width of the neutron, in electron volts?
(b) Classical resonances: Let us probe a bit what this ``width'' means, and what it has to do with decay rate. We'll start with a classical example: Consider a damped, driven harmonic oscillator:
x¨+Γx˙+ω02x=cosωt
Determine the frequency response of this classical oscillator, i.e., determine the square of the amplitude of oscillation as a function of ω. (Why the square?)
(c) In the limit of a narrow resonance, what is the full width at half maximum of the distribution you found in part b.
(d) Now suppose the driving force is turned off. How does the energy stored in the oscillator change with time? Find the ``lifetime'' of this oscillator. In this classical example, I mean the time it takes for the oscillator to reach e−1 of its original energy. Your answer should be very simply related to your answer for part c).