foregoing, it is reasonable to calculate the semi-classical single-electron spin current density from $$\label{eqn:scscd}
{\bf Q}_\pm(x,k_x)={\bf s}_\pm(x,k_x){\hbar k_x \over m}$$ and use the second equation in Eq. (\[eqn:spinandcurrent\]) to find ${\bf Q}_\pm(x)$. The correctness of this prescription is illustrated in Fig. \[fig:LLGcheck\](c).

Spin Spiral {#spinspiral}
===========

As a preliminary to our discussion of domain walls, it is instructive to discuss the spin density and spin current density for a spin spiral—an infinite magnetic structure where the direction of the magnetization rotates continuously as one moves along a fixed axis in space. Spin spirals occur in the ground state of some rare earth metals[@Jensen:1991] and also for the $\gamma$ phase of iron.[@Hafner:2002] Here, we focus on a spiral with uniform pitch $p$ where the magnetization rotates in the $x$-$z$ plane, [i.e.]{}, in Eq. (\[eqn:magnetization\]), $$\theta(x) = p\hspace{0.1em}x ~~~~~~~~~~{\rm and}~~~~~~~~~~\phi(x)=0. 
\label{thetas}$$ A cartoon version of this ${\bf M}(x)$ is shown in Fig. \[fig:spinspiral\]. This figure also defines a [*local*]{} coordinate system that will be useful in what follows. The system $(x',y',z')$ rotates as a function of $x$ so the magnetization ${\bf
M}(x)$ always points along $+z'$.

Calvo[@Calvo:1978] solved Eq. (\[eqn:Schrodinger\]) to find the eigenstates and eigen-energies of this spin spiral. In our 