By eqns. (11.3.11), the number operator \(\hat{n}_{\mathrm{p}}\) for the real particles is given by

\[
\hat{n}_{\mathrm{p}}=a_{\mathrm{p}}^{+} a_{\mathrm{p}}=\frac{1}{1+\alpha_{\mathrm{p}}^{2}}\left\{b_{\mathrm{p}}^{+} b_{\mathrm{p}}-\alpha_{\mathrm{p}}\left(b_{-\mathrm{p}} b_{\mathrm{p}}+b_{\mathrm{p}}^{+} b_{-\mathrm{p}}^{+}ight)+\alpha_{\mathrm{p}}^{2} b_{-\mathrm{p}} b_{-\mathrm{p}}^{+}ight\} .
\]

The terms linear in \(\alpha_{\mathrm{p}}\) do not contribute to the expectation value of \(\hat{n}_{\mathrm{p}}\) (with \(\mathbf{p} eq 0\) ) because of the absence of the diagonal matrix elements in \(b_{-\mathrm{p}} b_{\mathrm{p}}\) and \(b_{\mathrm{p}}^{+} b_{-\mathrm{p}}^{+}\). Further, since \(b_{\mathrm{p}}^{+} b_{\mathrm{p}}=\hat{N}_{\mathrm{p}}\) and \(b_{-\mathrm{p}} b_{-\mathrm{p}}^{+}=\hat{N}_{-\mathrm{p}}+1\), we get

\[
\overline{n_{\mathrm{p}}}=\frac{1}{1-\alpha_{\mathrm{p}}^{2}}\left\{\bar{N}_{\mathrm{p}}+\alpha_{\mathrm{p}}^{2}\left(\bar{N}_{-\mathrm{p}}+1ight)ight\} \quad(\mathbf{p} eq 0)
\]

Finally, in view of the isotropy of the problem, \(\bar{N}_{-\mathrm{p}}=\bar{N}_{\mathrm{p}}\) and we get the desired result.