2.5.1.2 Fermi's Golden Rule

Expression (2.94) cannot directly be used in the kinetic equation (2.60) because of two reasons. The first reason is that it relates to transitions between states of a discrete spectrum^2.31 while in the kinetic equation they are treated as continual states. The second one lies in the fact that in the semiclassical transport model scattering events are local in time. The first problem can be overcome by the fact that systems are usually macroscopic which allows to consider the discrete spectrum as quasi-continuous. The second problem is removed assuming that the free-flight time is much longer than the effective interaction time which allows to simplify (2.94) by taking the long interaction time limit^2.32, namely for $t\rightarrow\infty$ one gets:

$$\frac{d\vert a_{s^{'}}(t)\vert^{2}}{dt}=\frac{2\pi}{\hbar}\vert\langle s^{'}\vert H_\mathrm{int}\vert s\rangle\vert^{2}\delta(E_{s^{'}}-E_{s}).$$ (2.95)

This equality is called the Fermi golden rule and it shows that transitions only happen between states with equal energy $E_{s^{'}}=E_{s}$. The proof that expression (2.95) can be used in the kinetic equation (2.60) represents a difficult problem. For the example of impurity scattering such proof can be found in [22,23]. S. Smirnov: