Advanced Quantum Mechanics Schwabl · Problem 4.3

Relativistic Wave Equations and their Derivation

Physics — Quantum Mechanics · f-sum-rule-density-density

Problem

Prove the

f

-sum rule,

\int \frac{d\omega}{2\pi}\, \omega\chi''(\mathbf{k},\omega) = \int \frac{d\omega}{2\pi\hbar}\omega S(\mathbf{k},\omega) = \frac{k^2}{2m}n

for the density--density correlation function. Hint: Calculate

\frac{\mathrm{i}}{2\hbar}\langle[\dot{\rho}_{\mathbf{k}}, \rho_{-\mathbf{k}}]\rangle