# Bogoliubov superfluidity Hamiltonian

1. Oct 9, 2009

### Petar Mali

$$\hat{H}=\sum_{\vec{p}}\frac{p^2}{2m}\hat{b}^+_{\vec{p}}\hat{b}_{\vec{p}}+\frac{1}{2V}\sum_{\vec{p}_1,\vec{p}_2,\vec{p}_3}W(\vec{p}_1-\vec{p}_3)\hat{b}^+_{\vec{p}_1}\hat{b}^+_{\vec{p}_2}\hat{b}_{\vec{p}_3}\hat{b}_{\vec{p}_1+\vec{p}_2-\vec{p}_3}$$

Is this correct form or maybe?

$$\hat{H}=\sum_{\vec{p}}\frac{p^2}{2m}\hat{b}^+_{\vec{p}}\hat{b}_{\vec{p}}+\frac{1}{2N}\sum_{\vec{p}_1,\vec{p}_2,\vec{p}_3}W(\vec{p}_1-\vec{p}_3)\hat{b}^+_{\vec{p}_1}\hat{b}^+_{\vec{p}_2}\hat{b}_{\vec{p}_3}\hat{b}_{\vec{p}_1+\vec{p}_2-\vec{p}_3}$$

One form is in book and one in scripts! This is first relation so I'm not quite sure! Thanks for you're answer!

2. Oct 9, 2009

### hiyok

Actually both forms are correct. In the second, the author assumes the unit cell volumn is one, so N indicates the total volumn V.