Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S18 / 3

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.

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.

Evaluate the envelopes at their displayed arguments and set $a={\eta }^{\ast }\rho$. The common phase $e^{-i{V}_{0}t}$ cancels from every bilinear. Direct use of the Pauli matrices givesThe physical longitudinal current is $J_z=|\eta {|}^2-|\rho {|}^2$: the two density profiles move in opposite directions. Their separate transport identities give ${\partial }_t(|\eta {|}^2+|\rho {|}^2)+{\partial }_z(|\eta {|}^2-|\rho {|}^2)=0$.

For a coherent superposition, the transverse currents $J_x=2\mathrm{Re}({\eta }^{\ast }\rho )$ and $J_y=2\mathrm{Im}({\eta }^{\ast }\rho )$ generally do not vanish. They are independent of $x,y$, so they contribute no divergence in this one-dimensional construction. Omitting them would give an incorrect local four-current. Pointwise, $|\mathbf{J}{|}^2=(|\eta {|}^2+|\rho {|}^2{)}^2$, so the pure two-component Weyl current is future null wherever the density is nonzero. If only one envelope is present, the current points entirely along its propagation direction; an incoherent average can remove the transverse interference terms.