= Solution
In the supplied <Dirac representation of the gamma matrices>[Dirac representation], $\gamma^5$ exchanges the upper and lower blocks. Therefore
$$
P_L=\frac{1-\gamma^5}{2}=\frac12\begin{pmatrix}I&-I\\-I&I\end{pmatrix}.
$$
Define $\eta=[I-\boldsymbol\sigma\cdot\mathbf p/(E+m)]\chi$. Acting with this <Projection operator>[projector] gives
$$
\boxed{\psi_L=P_L\psi=\frac N2\begin{pmatrix}\eta\\-\eta\end{pmatrix}e^{i(\mathbf p\cdot\mathbf r-Et)}.}
$$
For completeness, its two independent components are
$$
\eta_1=\frac{\sqrt5}{3}\left(1-\frac{p_z}{E+m}\right)-\frac{2(p_x-ip_y)}{3(E+m)},\qquad
\eta_2=-\frac{\sqrt5(p_x+ip_y)}{3(E+m)}+\frac23\left(1+\frac{p_z}{E+m}\right).
$$
The equal-and-opposite block structure verifies $\gamma^5\psi_L=-\psi_L$. Projection isolates a <Chirality>[chiral component] of the massive <Dirac field>[Dirac] solution; that component alone generally does not satisfy an independent massive <Dirac equation>, since the <mass> couples both <chiralities>.
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