Advanced Quantum Mechanics Schwabl · Problem 2.2

Bosons

Physics — Quantum Mechanics · exchange-energy-electron-gas

Problem

Prove the validity of the following relations, which have been used in the evaluation of the energy shift

\Delta\epsilon(\mathbf{k})

of the electron gas, Eq. (2.2.21): a)

\displaystyle -4\pi e^2 \int \frac{d^3k'}{(2\pi)^3} \frac{1}{|\mathbf{k}-\mathbf{k}'|^2}\Theta(k_F - k') = -\frac{2e^2}{\pi}k_F F(k/k_F),

with

F(x) = \frac{1}{2} + \frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right|.

\displaystyle E^{(1)} = -\frac{e^2 k_F}{\pi}V\int\frac{d^3k}{(2\pi)^3}\left[1 + \frac{k_F^2 - k^2}{2kk_F}\ln\left|\frac{k_F+k}{k_F-k}\right|\right]\Theta(k_F - k)

\displaystyle = -\frac{3}{4}\frac{e^2 k_F}{\pi}N = -\frac{e^2}{2a_0 r_s}\left(\frac{9\pi}{4}\right)^{1/3}\frac{3N}{2\pi},

where

r_s

is a dimensionless number which characterizes the mean particle separation in units of the Bohr radius

a_0 = \hbar^2/me^2

. Furthermore,

k_F^3 = 3\pi^2 n = 1/(\alpha a_0 r_s)^3

with

\alpha = (4/9\pi)^{1/3}