Advanced Quantum Mechanics Schwabl · Problem 2.3

Bosons

Physics — Quantum Mechanics · hartree-fock-electron-gas

Problem

Apply the atomic Hartree–Fock equations to the electron gas. a) Show that the Hartree–Fock equations are solved by plane waves. b) Replace the nuclei by a uniform positive background charge of the same total charge and show that the Hartree term is canceled by the Coulomb attraction of the positive background and the electrons. The electronic energy levels are then given by ϵ(k)=(ℏk)22m−1V∑q4πe2∣k−q∣2Θ(kF−q).\epsilon(\mathbf{k}) = \frac{(\hbar\mathbf{k})^2}{2m} - \frac{1}{V}\sum_{\mathbf{q}}\frac{4\pi e^2}{|\mathbf{k}-\mathbf{q}|^2}\Theta(k_F - q). According to Problem 2.2, this can also be written as ϵ(k)=(ℏk)22m−2e2πkFF(k/kF)\epsilon(\mathbf{k}) = \frac{(\hbar k)^2}{2m} - \frac{2e^2}{\pi}k_F F(k/k_F).

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