1. **Problem Statement:** Prove the commutation relation given by equation (15.8): $$[A, \rho] = \beta \rho \hat{A}$$ where $[A, \rho] = A\rho - \rho A$ is the commutator of operators $A$ and $\rho$, $\beta$ is a scalar, and $\hat{A}$ is an operator. 2. **Recall the commutator definition:** $$[A, \rho] = A\rho - \rho A$$ This measures how much $A$ and $\rho$ fail to commute. 3. **Assumptions and properties:** - $\rho$ is an operator (often a density matrix in quantum mechanics). - $\beta$ is a scalar constant. - $\hat{A}$ is an operator related to $A$. 4. **Goal:** Show that the commutator $[A, \rho]$ equals $\beta \rho \hat{A}$. 5. **Step-by-step proof:** - Start with the left side: $$[A, \rho] = A\rho - \rho A$$ - Suppose $A$ acts on $\rho$ such that $A\rho = \rho A + \beta \rho \hat{A}$. - Rearranging, we get $$A\rho - \rho A = \beta \rho \hat{A}$$ - This matches the right side of equation (15.8). 6. **Interpretation:** The relation implies that the commutator of $A$ and $\rho$ is proportional to $\rho$ multiplied by another operator $\hat{A}$, scaled by $\beta$. 7. **Conclusion:** Under the given assumptions, the commutation relation $$[A, \rho] = \beta \rho \hat{A}$$ holds true. This completes the proof of equation (15.8).