diff options
Diffstat (limited to 'content/know/concept/jellium/index.pdc')
| -rw-r--r-- | content/know/concept/jellium/index.pdc | 7 |
1 files changed, 1 insertions, 6 deletions
diff --git a/content/know/concept/jellium/index.pdc b/content/know/concept/jellium/index.pdc index 15c3308..9743514 100644 --- a/content/know/concept/jellium/index.pdc +++ b/content/know/concept/jellium/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-11-23 categories: - Physics - Quantum mechanics +- Perturbation date: 2021-10-15T20:31:12+02:00 draft: false @@ -363,17 +364,11 @@ $$\begin{aligned} &= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F} \bigg[ \frac{k_F^3}{3} x + \frac{|\vb{q}|^3}{48 \xi^2} \bigg]_{|\vb{q}| / (2 k_F)}^1 \dd{|\vb{q}|} \\ - %&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F} - %\bigg( \frac{k_F^3}{3} + \frac{|\vb{q}|^3}{48} - \frac{k_F^3}{3} \frac{|\vb{q}|}{2 k_F} - \frac{4 k_F^2 |\vb{q}|^3}{48 |\vb{q}|^2} \bigg) - %\dd{|\vb{q}|} - %\\ &= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F} \bigg( \frac{k_F^3}{3} + \frac{|\vb{q}|^3}{48} - \frac{k_F^2 |\vb{q}|}{4} \bigg) \dd{|\vb{q}|} \\ &= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \bigg[ \frac{k_F^3 |\vb{q}|}{3} + \frac{|\vb{q}|^4}{192} - \frac{k_F^2 |\vb{q}|^2}{8} \bigg]_0^{2 k_F} \\ - %&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \bigg( \frac{k_F^3}{3} 2 k_F + \frac{16 k_F^4}{192} - \frac{k_F^2 4 k_F^2}{8} \bigg) - %\\ &= \frac{- e^2 V}{16 \pi^4 \varepsilon_0} k_F^4 = \frac{- e^2 N}{16 \pi^4 \varepsilon_0 n} k_F^4 = -\frac{3 e^2 N}{16 \pi^2 \varepsilon_0} k_F |