For the last step in the derivation of the Gross-Pitaevskii equation, we have the following equation(adsbygoogle = window.adsbygoogle || []).push({});

[itex]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,[/itex]

where [itex]\eta[/itex] is an arbitrary function, [itex]g,N,\mu[/itex] are constants, [itex]h[/itex] is the hamiltonian for the harmonic oscillator and [itex]\phi[/itex] is the ground state of the hamiltonian.

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

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

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