Rewrite the Gross-Pitaevskii equation and the mean field energy, see equations (11.2.21) and (11.2.23), for an isotropic harmonic oscillator trap with frequency \(\omega_{0}\) in a dimensionless form by defining a dimensionless wavefunction \(\psi=a_{\mathrm{Osc}}^{3 / 2} \Psi / \sqrt{N}\), a dimensionless length \(s=r / a_{\mathrm{osc}}\), and a dimensionless energy \(E / N \hbar \omega_{0}\). Show that the dimensionless parameter that controls the mean field energy is \(N a / a_{\mathrm{osc}}\), where \(N\) is the number of particles in the condensate, \(a\) is the scattering length, and \(a_{\mathrm{osc}}=\sqrt{\hbar / m \omega_{0}}\). Next, show that the dimensionless versions of the Gross-Pitaevskii equation and the mean field energy are

\[-\frac{1}{2} \tilde{abla}^{2} \psi+\frac{1}{2} s^{2} \psi+\frac{4 \pi N a}{a_{\text {osc }}}|\psi|^{2} \psi=\tilde{\mu} \psi\]

and

\[\frac{E[\psi]}{N \hbar \omega_{0}}=\int\left(\frac{1}{2}|\tilde{abla} \psi|^{2}+\frac{1}{2} s^{2}|\psi|^{2}+\frac{2 \pi N a}{a_{\text {osc }}}|\psi|^{4}\right) d s\]