Relativistic electron in a local, central potential

Much can be learnt from (4.166). First we can use it to interpret the resonance at Er physically. The limit 3 —y aa corresponds to a sudden excitation of the target in comparison with times characteristic of the target, in this case 

The time spectrum of the scattered electron in this limit has the shape g-Frf/fi jjjg lifetime of the compound electron—target system in the resonant state is Xr. We have derived the same result for a detailed scattering theory that we knew already from the uncertainty principle. The uncertainty relation (4.146) becomes an exact equality (4.167) if Tr is the full width at half maximum and Xr is the lifetime. 

The differential cross section, obtained by integrating (r, t)p, given by (4.166), over time, is 

In the usual scattering experiment 5 — 0 and (4.168) reduces to the Breit-Wigner form (4.143). Equn. (4.168) also tells us under what conditions the wave-packet width is significant in the experiment. We must have experimental time resolution H/3 such that 5 is comparable to the resonance width Tr- The width in this case is the sum of Tr and 3. 

While we have developed the theory of wave-packet scattering and resonances in the context of potential scattering of electrons it is easy to generalise. In particular there is no reason why the scattered particle should not be a photon. In this case the wave packet does not spread and the formalism is valid for general values of 3. Wave packets are known whose widths correspond to a lifetime of order lO s, which is easily resolved with nanosecond electronics. Such wave packets arise in the photon decay of many atomic states. The time spectrum of detected photons is given by (r,t)p for X 0. We see from (4.166) that this involves an interference between a term whose lifetime is h/3 and one whose lifetime is Xr. The resulting time oscillations have been observed experimentally. They are called quantum beats. 

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