Bogoliubov superfluidity Hamiltonian

[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!

 

[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.

 

[charng]

The Bogoliubov superfluidity Hamiltonian is a commonly used model in the study of superfluids and Bose-Einstein condensates. It describes the energy of a system of interacting bosons in a superfluid state, where the bosons are represented by creation and annihilation operators. The first term in the Hamiltonian represents the kinetic energy of the bosons, while the second term represents the interaction energy between them. The form of the second term may vary depending on the specific system or context. It is important to consult reliable sources and experts in the field for the correct form of the Hamiltonian in a particular case.