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?