nLab Breit-Wigner distribution

Due to decay, an unstable quantum-mechanical system cannot be in the eigenstate of the Hamiltonian. Effectively, this means that the system is described by a wave-packet which ranges over a distribution of energies (or frequencies, as in the field of spectroscopy). In the simplest model of an exponential decay, the distribution of energies is given by so called Breit-Wigner distribution (and to some extent this is true for most short-living metastable system, e.g. resonances in high energy physics).

Given some quantities $E_0$ and $\Gamma$ of the dimension of energy the Breit-Wigner energy distribution

$$|\rho(E)|^2 = \frac{\Gamma/2\pi}{\frac{1}{2}\Gamma^2+(E-E_0)^2}$$

which should be in principle damped all the way to $0$ from some energy $E_{min}$ as the spectrum of any Hamiltonian must be bounded below. The plot of the Breit-Wigner curve as seen in spectroscopy is often called Breit-Wigner curve.

Various effects in nature lead to modifed, asymmetric and other spectral curves to which the Breit-Wigner is merely the crudest approximation.

Literature

Classical Breit-Wigner

• Anthony Sudbery, Quantum mechanics and the particles of nature, formula 3.237 and around.
• G. Breit, E. Wigner, Phys. Rev. 49, 519 (1936).

Corrected curves

Assymetric line shapes in autoionization and Rydberg atoms are studied in the famous

• Ugo Fano, Effects of configuration interaction on intensities and phase shifts, Phys. Rev. 124, 1866 (1961) doi
• wikipedia: Fano resonance
• A. Bianconi, Ugo Fano and shape resonances in X-ray and Inner Shell Processes AIP Conference Proceedings (2002): (19th Int. Conference Roma June 24–28, 2002) cond-mat/0211452
• VV Sokolov, VG Zelevinsky, Dynamics and statistics of unstable quantum states, Nuclear Physics A504:3 (1989) 562-588 doi; Simple mode on a highly excited background: Collective strength and damping in the continuum, Phys. Rev. C 56, 311 (1997)

category: physics