Show that in an interacting Bose gas the mean occupation number \(\bar{n}_{\boldsymbol{p}}\) of the real particles and the mean occupation number \(\bar{N}_{p}\) of the quasiparticles are connected by the relationship

\[\bar{n}_{\boldsymbol{p}}=\frac{\bar{N}_{\boldsymbol{p}}+\alpha_{\boldsymbol{p}}^{2}\left(\bar{N}_{\boldsymbol{p}}+1\right)}{1-\alpha_{\boldsymbol{p}}^{2}} \quad(\boldsymbol{p} eq 0)\]

where \(\alpha_{\boldsymbol{p}}\) is given by equations (11.3.9) and (11.3.10). Note that equation (11.3.22) corresponds to the special case \(\bar{N}_{\boldsymbol{p}}=0\).