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 electromagnetic 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 g1, we can
write the time dependent wave function as (h=1)
We
can now express this wave in terms of an orthonormal basis set for the
time dependence of stationary solutions of a Hamilton operator.
Where
the coefficients a(E) are projection of the wave function on the basis
set functions. They are given by the scalar product.
Solving this integral using 0 and infinity as limits we obtain:
The probability density of finding the particle in a state with energy E is now:
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; 5473
 F.Hoyt; Phys. Rev,; 36, 1930, 860870
 W. Heitler, S.T. Ma, Proc. R.I.A. (A); 52; 1949, 109125
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.
