Proof of the conservation of the energy functional for the Gross-Pitaevskii equation?

From the Gross-Pitaevskii equation \begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V+g|\psi|^2\right)\psi\end{equation} using the variational relation \begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\frac{\partial\varepsilon}{\partial \psi^*}\end{equation} we find the energy density \begin{equation}\varepsilon=\frac{\hbar^2}{2m}|\nabla\psi|^2+V|\psi|^2+\frac{g}{2}|\psi|^4\end{equation} The energy would be $E=\int d^3r \varepsilon$ and this is a prime integral of the motion, meaning it is a conserved quantity.

My questions are:

1) How do we get the variational relation?

2)How can we prove that $E$ is a conserved quantity?

• 1) You may read the chapter 7 of Pethick and Smith's book on BEC. 2) $\epsilon$ is energy density which is not conserve, only the total energy – unsym Jul 9 '14 at 23:24
• 1) Even if I read that book already I didn't remember that. Thank you. 2) Of course you are right, my mistake; I will amend the question – Semola Jul 9 '14 at 23:37
• Do you know a nice way to show the conservation of $E$ different from brute force computation? – Semola Jul 9 '14 at 23:41
• After an integration by parts, $\epsilon$ is almost the operator that gives the time-evolution, and one could perhaps try to copy the proof of the Ehrenfest theorem. – Robin Ekman Jul 10 '14 at 0:35
• I think it won't work because the exponential would depend on $|\psi|^2$ as well. – Semola Jul 11 '14 at 13:40

by computing $\partial_t E$ and using what given in the Gross-Pitaevskii for $\dot{\psi}$ and $\dot{\psi^*}$ one can check that all the terms cancel out so that $\partial_tE=0$.
• Do you think Dirac-Frenkel variational principle (which can be derived as a saddle point approximation for the coherent state Feynman path integral) with the wave functional ansatz $$|\{\psi(\bar{r},t)\}>=e^{\int_{V}d^3_{}\bar{r}\psi(\bar{r},t)\Psi_{}^{\dagger}(\bar{r})}|\ \text{vac}>$$ will lead to the Gross-Pitaevskii equation? – Sunyam Oct 6 '17 at 15:16