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Gross-Pitaevskii equation

In order to derive the equation for the wavefunction of the condensate $\Phi({\bf r},t)$ one has to write the time evolution of the field operator $\hat\Psi({\bf r},t)$ using the Heisenberg equation
$\displaystyle i \hbar \frac{\partial}{\partial t} \hat\Psi({\bf r},t)
= [\hat\Psi, \hat H]$     (1.9)

Substitution of the Hamiltonian (1.3) into (1.9) gives
$\displaystyle i \hbar \frac{\partial}{\partial t} \hat\Psi({\bf r},t)
=\left( -...
...ngle + V({\bf r})
+g \vert\hat\Psi({\bf r},t)\vert^2\right) \hat\Psi({\bf r},t)$     (1.10)

We now replace the field operator $\hat\Psi({\bf r},t)$ with the classical field $\Phi({\bf r},t)$. Then, the following equation for the order parameter is obtained
$\displaystyle i \hbar \frac{\partial}{\partial t} \Phi({\bf r},t)
=\left( -\fra...
...{2m}\triangle + V({\bf r})
+g\vert\Phi({\bf r},t)\vert^2 \right)\Phi({\bf r},t)$     (1.11)

This equation is called Gross-Pitaevskii equation [12,13,14] and describes the time evolution of the order parameter.q