Advanced Quantum Mechanics Schwabl · Problem 2.1

Bosons

Physics — Quantum Mechanics · static-structure-function-noninteracting-fermions

Problem

Calculate the static structure function for noninteracting fermions

S^0(\mathbf{q}) \equiv \frac{1}{N}\langle\phi_0 | \hat{n}_{\mathbf{q}}\hat{n}_{-\mathbf{q}} | \phi_0\rangle ,

where

\hat{n}_{\mathbf{q}} = \sum_{\mathbf{k},\sigma} a^\dagger_{\mathbf{k}\sigma}a_{\mathbf{k}+\mathbf{q}\sigma}

is the particle density operator in the momentum representation and

|\phi_0\rangle

is the ground state. Take the continuum limit

\sum_{\mathbf{k},\sigma} = 2V\int d^3k/(2\pi)^3

and calculate

S^0(\mathbf{q})

explicitly. Hint: Consider the cases

\mathbf{q} = 0

and

\mathbf{q} \neq 0

separately.