Breit-Wigner equation

As a result, the energy E of photons emitted by an ensemble of identical nuclei, rigidly fixed in space, upon transition from their excited states (e) to their ground states (g), scatters around the mean energy Eq = E. Eg. The intensity distribution of the radiation as a function of the energy E, the emission line, is a Lorentzian curve as given by the Breit-Wigner equation [1] ... 

Equations (15) and (16) are Breit-Wigner s one-level formula for the phase shift. If the pole lies close to the real E axis, i.e., if T is very small, the part 5r of the phase shift increases very rapidly with E by tt/2 within the energy region of width T and centered at Er. It increases by nearly it within several times T. This is a resonance phenomenon. 

The Breit-Wigner approximation allows one to represent the resonance tunneling spectrum as superposition of Lorentzian lines. In the limit of narrow electronic levels I o 0- the resonance transparency obeys the equation... 

In the multichannel case, the quantity that follows the Breit-Wigner form of Equation 1.63 is the S-matrix eigenphase sum [38,39], which is the sum of the phases obtained from the eigenvalues of the S-matrix. The eigenphases and the eigenphase sum are real, unlike the phases S obtained from individual diagonal elements, because the S-matrix is unitary, so that all its eigenvalues have modulus 1. 

The position and width of the resonance can be easily related to the derivatives with respect to k. Assuming the Breit-Wigner expression for the resonant part of the phase shift, the following equations result ... 

This equation gives the very important fact that the energy of decaying state is not a constant and is distributed over a region with a width determined by the decay constant. The width is called natural line width. The shape of the distribution is called a Lorentzian or Breit-Wigner curve as shown in Fig. 1.3. 

The Breit-Wigner formula points to an interesting effect of having a molecular level lie exactly at the Fermi energy. When 0 Equation 6.4, the formula reduces to... 

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