Breit-Wigner Distribution

  Probability density functions of the general form are also known in statistics as Cauchy distributions. The Breit-Wigner (also known as Lorentz) distribution is a generalized form originally introduced ([Breit36], [Breit59]) to describe the cross-section of resonant nuclear scattering in the form

which had been derived from the transition probability of a resonant state with known lifetime. The equation follows from that of a harmonic oscillator with damping, and a periodic force.

The above form can be read as the definition of a probability density as a function of E, the integral over all energies E is 1. Variance and higher moments of the Breit-Wigner distribution are infinite. The distribution is fully defined by E₀, the position of its maximum ( about which the distribution is symmetric), and by , the full width at half maximum (FWHM), as obviously

The Breit-Wigner distribution has also been widely used for describing the non-interfering cross-section of particle resonant states, the parameters E₀ (= mass of the resonance) and (= width of the resonance) being determined from the observed data. Observed particle width distributions usually show an apparent FWHM larger than , being a convolution with a resolution function due to measurement uncertainties. and the lifetime of a resonant state are related to each other by Heisenberg's uncertainty principle ( ).

A normal (Gaussian) distribution decreases much faster in the tails than the Breit-Wigner curve. For a Gaussian, FWHM = 2.355 , [ here is the distribution's standard deviation]. The Gaussian in the graph above would be even more peaked at x = 0 if it were plotted with FWHM equal to 1 (as the Breit-Wigner curve).