1. **Stating the problem:** We want to derive equations (15.3) and (15.4) which describe the scattering time $\tau_0(kk')$ and the transition rate $W(kk')$ in terms of the energies and perturbations in the system. 2. **Starting point:** Equation (15.2) involves a $\delta$-function representing energy conservation and a sum over initial states weighted by $\rho_I = \exp\{\beta(\Omega - E_I)\}$. 3. **Using the $\delta$-function as a time integral:** Recall the identity $$\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dt \, e^{ixt}$$ Applying this to the energy difference $\epsilon_{sk} - \epsilon_{s k'}$ in the $\delta$-function allows us to write the transition probability as a time integral of an exponential factor. 4. **Unperturbed average over electrons:** Taking the average over electron states without perturbation simplifies the expression, while the perturbation effects are included via the Boltzmann equation (13.6). 5. **Deriving equation (15.3):** The scattering time $\tau_0(kk')$ is expressed as $$\tau_0(kk') = W(kk') f_0(\epsilon_{sk}) [1 - f_0(\epsilon_{s k'})]$$ where $f_0$ is the Fermi-Dirac distribution function. This formula accounts for the probability of occupation and vacancy of initial and final states respectively. 6. **Deriving equation (15.4):** The transition rate $W(kk')$ is given by $$W(kk') = \hbar^{-1} \sum_q |\alpha_{k\mu}|^2 \delta_{k', k-q} \int dt \, e^{i(\epsilon_{sk} - \epsilon_{s k'})t} h(q; t) h^+(q; 0)$$ This expression sums over momentum transfer $q$, includes the matrix element $|\alpha_{k\mu}|^2$, enforces momentum conservation via $\delta_{k', k-q}$, and integrates over time the correlation functions $h(q; t)$ and $h^+(q; 0)$ weighted by the energy difference exponential. 7. **Summary:** By rewriting the $\delta$-function as a time integral and averaging over unperturbed electron states, we arrive at the expressions (15.3) and (15.4) that relate scattering time and transition rates to the microscopic perturbations and electron distributions. **Final answers:** $$\tau_0(kk') = W(kk') f_0(\epsilon_{sk}) [1 - f_0(\epsilon_{s k'})]$$ $$W(kk') = \hbar^{-1} \sum_q |\alpha_{k\mu}|^2 \delta_{k', k-q} \int dt \, e^{i(\epsilon_{sk} - \epsilon_{s k'})t} h(q; t) h^+(q; 0)$$