LorentzFunktionen

LorentzFunktionen, in English Lorentzian functions, describe a family of peak-shaped functions used to model resonances and spectral lines. They are centered at x0 and characterized by a width parameter Γ, which is often interpreted as the full width at half maximum (FWHM).

A common form is L(x; x0, Γ) = (1/π) · ((Γ/2) / ((x − x0)² + (Γ/2)²)). Multiplying by an amplitude A yields L_A(x) = A · L(x; x0, Γ). When normalized to unit area, the Lorentzian is a probability density function of the Cauchy distribution with location x0 and scale γ = Γ/2. The Lorentzian has heavy tails, and, in the distribution sense, its mean and variance are not defined.

In the time and frequency domains, the Lorentzian is linked to exponential decay. Its Fourier transform is F{L}(ω) = e^(−i ω x0) · e^(−(Γ/2) |ω|), reflecting its connection to damped oscillations. The width Γ is related to lifetimes in quantum systems and to broadening mechanisms in spectroscopy.

Applications of Lorentzfunktionen include modeling natural linewidths in atomic and molecular spectroscopy, resonances in particle and condensed-matter physics (where the Breit–Wigner distribution is a relativistic form of the same shape), and fitting spectral profiles. In practice, Lorentzians are often convolved with Gaussians to form Voigt profiles, which account for both natural broadening and Doppler broadening. The term LorentzFunktionen is widely used in German-language physics literature to denote these peak-shaped functions.