 Solution

For either normalized two-spinor ${\chi }^s$, $(\mathbf{\sigma }\cdot \mathbf{p}{)}^2=p^{2}I$ givesThe Dirac equation alone does not fix $N$: a normalization convention is needed. In the usual covariant external-spinor convention ${\bar{u}}_r{u}_s=2m{\delta }_{rs}$, choose real positive $N=\sqrt{E+m}$. Then $u_r^{†}{u}_s=2E{\delta }_{rs}$ and ${\bar{u}}_s{\gamma }^{\mu }{u}_s=2p^{\mu }$. A second common convention is unit rest density, $\bar{u}u=1$, giving $N=\sqrt{(E+m)/(2m)}$ and density $E/m$. Both explain why the boosted density is not one particle per unit coordinate volume: the density is a four-current component and changes under a Lorentz transformation.

One may instead choose unit laboratory density, $N=\sqrt{(E+m)/(2E)}$, or box normalization $N=\sqrt{(E+m)/(2EV)}$ for one particle in volume $V$. An infinite plane wave is not normalizable to one particle over all space; a localized wave packet is required for that. Thus a unique $N$ cannot be inferred from the prompt without specifying which of these conventions is intended.