 Solution

Assume the scalar potential is real, so the Weyl equation has a Hermitian Hamiltonian $H=\mathbf{\sigma }\cdot \mathbf{p}+V$. Multiplying the equation and its adjoint by ${\psi }^{†}$ and $\psi$ respectively givesThus with $\rho ={\psi }^{†}\psi$ and $\mathbf{J}={\psi }^{†}\mathbf{\sigma }\psi$, the lower-index four-current $j_{\mu }=(\rho ,-\mathbf{J})$ obeys $\boxed{{\partial }^{\mu }{j}_{\mu }=0}$.

For $K=\mathbf{\sigma }\cdot \mathbf{p}$,Consequently $K$ is conserved for spatially constant $V$; a spatially uniform time-dependent scalar offset would also commute with it. For general nonuniform $V$ it is not conserved. At nonzero momentum, $(\mathbf{\sigma }\cdot \hat{\mathbf{p}}{)}^2=I$, so the helicity eigenvalues are $\boxed{+1,-1}$. In the chosen constant-potential Weyl Hamiltonian, the positive $E-V$ branch has helicity $+1$; the negative branch has $-1$. An expectation value in a superposition need not itself be an eigenvalue.