2.3.2.2 Pauli Exclusion Principle

The quantity $S(\vec{k},\vec{k}^{'},\vec{r},t)d\vec{k}^{'}$ represents the probability per unit time that an electron with wave vector $\vec{k}$ will be scattered to one of the levels in the domain $d\vec{k}^{'}$ around $\vec{k}^{'}$ if the levels are not occupied. Thus the real rate of transitions should be less than this quantity by a factor given by the ratio between available levels and the total number of levels, as the Pauli principle forbids transitions to occupied levels. This is schematically shown in Fig. 2.5.

Figure 2.5: The Pauli exclusion principle forbids transitions to the states which are already occupied by electrons.

The relative number of available states is equal to $1-f(\vec{k}^{'})$. The total probability that an electron will be scattered is given by the sum over all $\vec{k}^{'}$, which can be converted to an integration over $\vec{k}^{'}$:

$$\int (1-f(\vec{r},\vec{k}^{'},t))S(\vec{k},\vec{k}^{'},\vec{r},t)\,d\vec{k}^{'}.$$ (2.49)

S. Smirnov: