How/when can I take a wave function and its complex conjugate as independent?

[SiggyYo]

For the last step in the derivation of the Gross-Pitaevskii equation, we have the following equation
$0=\int \eta^*(gNh\phi+gN^2\phi^*\phi^2-N\mu\phi)\ dV+\int (N\phi^*h+gN^2(\phi^2)^*\phi-N\mu\phi^*)\eta\ dV,$
where $\eta$ is an arbitrary function, $g,N,\mu$ are constants, $h$ is the hamiltonian for the harmonic oscillator and $\phi$ is the ground state of the hamiltonian.

Now, the last step involves seeing that this can only be the case if $gNh\phi+gN^2\phi^*\phi^2-N\mu\phi$ and $N\phi^*h+gN^2(\phi^2)^*\phi-N\mu\phi^*$ are both zero. As far as I can tell, I would need an argument that $\phi$ and $\phi^*$ are independent for this to be true.

Can anyone explain why this is the case (or in case I'm wrong, explain what else I need to consider)?

Thanks,

 

[Bill_K]

Write it as ∫ η* A dV + ∫ η B dV = 0, and let η = α + i β where α, β are arbitrary real functions. Then ∫ [α (A + B) + β (-iA + iB)] dV = 0. This can only happen if A + B and A - B are both zero, in other words A and B are both zero. You don't have to assume anything about φ.