 Solution

For dependence on $z$ alone and constant $V_0$, remove the common phase by $\psi =e^{-i{V}_{0}t}\tilde{\psi }$. Then ${\partial }_t\tilde{\psi }=-{\sigma }_z{\partial }_z\tilde{\psi }$. The normalized eigenvectorsobey ${\sigma }_z{u}_{\pm }=\pm u_{\pm }$. The corresponding first-order transport equations have characteristics $t-z$ and $t+z$, respectively, givingHere $\eta$ 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 $(\mathbf{\sigma }\cdot \hat{\mathbf{n}})u_{\pm }(\hat{\mathbf{n}})=\pm u_{\pm }(\hat{\mathbf{n}})$. If $\hat{\mathbf{n}}=(\sin \vartheta \cos \phi ,\sin \vartheta \sin \phi ,\cos \vartheta )$, one choice isIndependent constant phases on the spinors can be absorbed into the two arbitrary envelope functions.