= Solution
For dependence on $z$ alone and constant $V_0$, remove the common phase by $\psi=e^{-iV_0t}\widetilde\psi$. Then $\partial_t\widetilde\psi=-\sigma_z\partial_z\widetilde\psi$. The normalized eigenvectors
$$
\boxed{u_+=\begin{pmatrix}1\\0\end{pmatrix},\qquad u_-=\begin{pmatrix}0\\1\end{pmatrix}}
$$
obey $\sigma_zu_\pm=\pm u_\pm$. The corresponding first-order transport equations have characteristics $t-z$ and $t+z$, respectively, giving
$$
\boxed{\psi(z,t)=e^{-iV_0t}\left[\eta(t-z)u_++\rho(t+z)u_-\right].}
$$
Here $\eta$ propagates toward $+z$ and $\rho$ toward $-z$, both at unit speed.
For a fixed unit vector $\hat{\mathbf n}$, replace $z$ by $\hat{\mathbf n}\cdot\mathbf x$ and choose orthonormal spinors satisfying $(\boldsymbol\sigma\cdot\hat{\mathbf n})u_\pm(\hat{\mathbf n})=\pm u_\pm(\hat{\mathbf n})$. If $\hat{\mathbf n}=(\sin\vartheta\cos\varphi,\sin\vartheta\sin\varphi,\cos\vartheta)$, one choice is
$$
u_+(\hat{\mathbf n})=\begin{pmatrix}\cos(\vartheta/2)\\ e^{i\varphi}\sin(\vartheta/2)\end{pmatrix},\qquad
u_-(\hat{\mathbf n})=\begin{pmatrix}-e^{-i\varphi}\sin(\vartheta/2)\\\cos(\vartheta/2)\end{pmatrix}.
$$
Independent constant phases on the spinors can be absorbed into the two arbitrary envelope functions.
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