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I am reading an article about Gross-Pitaevskii Equation using a variational method approach. I am confused about a step of the derivation of free energy.

We want to minimize:$$F=E-\mu N$$ and $$E(\psi)=\frac{\langle\psi|\hat{H}| \psi\rangle}{\langle\psi \mid \psi\rangle}$$

Using a mean-field approximation: $$|\Psi\rangle=|\psi\rangle \otimes|\psi\rangle \otimes \cdots \otimes|\psi\rangle$$

then the quantity we want to minimize is:$$F(\Psi)=\langle\Psi|\hat{H}| \Psi\rangle-\mu\langle\Psi \mid \Psi\rangle$$ so each wavefunction is identitcal.

My question is: how do we get the last equation? I know that when considering a product space wavefunction, the energy eigenvalue is additive for each wavefunction. Is it because of this, we can eliminate the $N$ in the equation?

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The number of particles, N, in the equation is related to the wavefunction by:

\begin{equation} \langle\Psi \mid \Psi\rangle = \int|\Psi(\mathbf{r})|^{2} d^{3} r = N \end{equation}

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