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Lorentz lineshape

In the classical picture, interpreting the photon emission as a damped oscillation, a Fourier analysis of the oscillation leads to a Lorentz shape. This can be found in most text books on the subject, i.e. Anne Thorne's book on spectroscopy.
Spontaneous photon emission by an atom in an excited that, can be explained by the coupling of the atom to the quantised electro-magnetic field. Treating the interaction as a first order perturbation leads to both the exponential decay behaviour and Lorentz shape in the emitted photon distribution. The coupling also introduces a shift in the energy levels, which in most spectroscopy text books is treated in the chapter on pressure broadening and pressure shift.
The following is no attempt to derive the full theory again, it only intends to demonstrate the apparent link between Fourier transform, the exponential decay of states and the Lorentzian line shape.
If we assume a particle in the state A with energy EA and we further assume the probability of finding the particle in the state A to decrease exponentially with the characteristic decay time g-1, we can write the time dependent wave function as (h=1)wave function of decaying state


We can now express this wave in terms of an orthonormal basis set for the time dependence of stationary solutions of a Hamilton operator.Expansion of wave function


Where the coefficients a(E) are projection of the wave function on the basis set functions. They are given by the scalar product.

energy coefficient of decaying wave function

Solving this integral using 0 and infinity as limits we obtain:Energy coefficient of Fourier transformation


The probability density of finding the particle in a state with energy E is now:Energy probability distribution of decaying state


Which is pretty close to the Lorentz shape of the spectral lines observed and the presence of the integrals of the typical Fourier form is quite obvious. This uncertainty in the energy of an unstable state also ensures the conservation of energy, when photons with a Lorentzian energy spread are emitted.
As usual a proper derivation is a little more complex, and can be found in the following references.

References:

  • V. Weisskopf, E. Wigner, Z. f. Phys.; 63; 1930; 54-73
  • F.Hoyt; Phys. Rev,; 36, 1930, 860-870
  • W. Heitler, S.T. Ma, Proc. R.I.A. (A); 52; 1949, 109-125

Reference books:

  • A.Messiah, Mécanique Quantique, tome 2, Dunod, 2003.
  • Anne P. Thorne, Spectrophysics; Chapman and Hall Ltd; 2nd ed. 1988
  • Robert D. Cowan, The Theory of Atomic Structure and Spectra; University of California Press; 1981

First published on the web: 15. 02. 2008.

Author: Thomas Nelis. The text is an extension of the discussion following Prof. E.B Steers lecture during the first GLADNET Training course held in Antwerp, Be, in September 2007.