1. The problem involves understanding the expression for the transition rate $\Gamma_0(kk')$ given by equation (15.3) and the detailed form of $W(kk')$ in equation (15.4). 2. Equation (15.3) states: $$\Gamma_0(kk') = W(kk') f_0(\epsilon_{sk}) [1 - f_0(\epsilon_{sk'})]$$ where $f_0(\epsilon)$ is the Fermi-Dirac distribution function, representing the probability that a state with energy $\epsilon$ is occupied. 3. Equation (15.4) defines $W(kk')$ as: $$W(kk') = \hbar^{-1} \sum_q |\alpha_{k',k}|^2 \delta(\epsilon_{sk'} - \epsilon_{sk} - q) \int dt e^{i(\epsilon_{sk} - \epsilon_{sk'})t/\hbar} \langle h(q;t) h^+(q;0) \rangle$$ 4. Here, $\epsilon_{sk}$ is the energy of the s-band electron with wavevector $k$, $\alpha_{k',k}$ is the matrix element of the perturbation, and the delta function enforces energy conservation. 5. The integral over time $t$ and the correlation function $\langle h(q;t) h^+(q;0) \rangle$ represent the dynamics of the perturbation in the system. 6. The problem essentially shows how the transition rate $\Gamma_0(kk')$ can be expressed in terms of the unperturbed electron states and the perturbation treated via the Boltzmann equation. 7. This formulation is important in quantum transport and scattering theory, linking microscopic interactions to observable transition rates. Final answer: $$\boxed{\Gamma_0(kk') = W(kk') f_0(\epsilon_{sk}) [1 - f_0(\epsilon_{sk'})]}$$ with $$W(kk') = \hbar^{-1} \sum_q |\alpha_{k',k}|^2 \delta(\epsilon_{sk'} - \epsilon_{sk} - q) \int dt e^{i(\epsilon_{sk} - \epsilon_{sk'})t/\hbar} \langle h(q;t) h^+(q;0) \rangle$$