I need to find the approximate solution of nonlinear Schrodinger equation $$ i\hbar \partial_{t} \Psi + \frac{\hbar^{2}}{2m}\Delta \Psi - g |\Psi|^{2}\Psi - \frac{m\omega^2 (x^2 + y^2 + z^2)}{2}\Psi = 0 $$ by using variational method. $\Psi$ has norm $\int |\Psi|^{2}d^{3}\mathbf r = N$, where N is the number of particles.

It was saying to me that I must start from the trial function $$ \Psi (x, y, z) = h e^{-\frac{1}{2\omega^{2}}\left( x^2 + y^2 + z^2\right)}e^{\frac{i}{\hbar}\mu t}, $$

where $\mu$ plays role of chemical potential.

So by using the property of $\Psi$ I got $h = \frac{\sqrt{N}}{(\omega \pi )^{\frac{3}{4}}}$. Then, due to the variational method, I must compute functional $J(N) = \langle \Psi | \hat {H}| \Psi\rangle$ and determine N from relation $\frac{\partial J (N)}{\partial N} = 0$. But I don't understand this step, because $N$ already is given in the task and equal to the integer number. So I don't have parameters in the trial function which can minimize functional $J(N)$.

Can you help me? Maybe, there is the mistake in the trial function and it must be some parameter $l^2$ instead of $\omega^{2}$?

This post imported from StackExchange Physics at 2014-03-21 17:05 (UCT), posted by SE-user Andrew McAddams