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Bosons
Physics — Quantum Mechanics · dynamic-structure-function-noninteracting-fermions
Problem Show that, for noninteracting fermions,
S 0 ( q , ω ) ≡ 1 N ∫ − ∞ + ∞ d t e i ω t ⟨ ϕ 0 ∣ n ^ q ( t ) n ^ − q ( 0 ) ∣ ϕ 0 ⟩ = ℏ V 2 π 2 N ∫ d 3 k Θ ( k F − k ) Θ ( ∣ k + q ∣ − k F ) × δ ( ℏ ω − ℏ 2 2 m ( q 2 + 2 k ⋅ q ) ) . S^0(\mathbf{q},\omega) \equiv \frac{1}{N}\int_{-\infty}^{+\infty} dt\, e^{i\omega t}\langle\phi_0|\hat{n}_{\mathbf{q}}(t)\hat{n}_{-\mathbf{q}}(0)|\phi_0\rangle
= \frac{\hbar V}{2\pi^2 N}\int d^3k\,\Theta(k_F - k)\,\Theta(|\mathbf{k}+\mathbf{q}| - k_F)
\times\delta\!\left(\hbar\omega - \frac{\hbar^2}{2m}(q^2 + 2\mathbf{k}\cdot\mathbf{q})\right). S 0 ( q , ω ) ≡ N 1 ∫ − ∞ + ∞ d t e iω t ⟨ ϕ 0 ∣ n ^ q ( t ) n ^ − q ( 0 ) ∣ ϕ 0 ⟩ = 2 π 2 N ℏ V ∫ d 3 k Θ ( k F − k ) Θ ( ∣ k + q ∣ − k F ) × δ ( ℏ ω − 2 m ℏ 2 ( q 2 + 2 k ⋅ q ) ) . ∫ − ∞ + ∞ d ω 2 π S 0 ( q , ω ) = { N for q = 0 , 1 − 1 N ∑ k , σ n k σ n k + q σ for q ≠ 0 \int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}S^0(\mathbf{q},\omega) = \begin{cases} N & \text{for } \mathbf{q} = 0\ , \\ 1 - \frac{1}{N}\sum_{\mathbf{k},\sigma} n_{\mathbf{k}\sigma}n_{\mathbf{k}+\mathbf{q}\sigma} & \text{for } \mathbf{q} \neq 0 \end{cases} ∫ − ∞ + ∞ 2 π d ω S 0 ( q , ω ) = { N 1 − N 1 ∑ k , σ n k σ n k + q σ for q = 0 , for q = 0 n ^ k σ = a k σ † a k σ \hat{n}_{\mathbf{k}\sigma} = a^\dagger_{\mathbf{k}\sigma}a_{\mathbf{k}\sigma} n ^ k σ = a k σ † a k σ Browse the problem library