brackets in Eq. (\[eqn:spinandcurrent\]).[@linearBoltzmann] The lines in Fig. \[fig:equilibrium\](a) are the Cartesian components of the imposed ${\bf M}(x)$. The solid dots in Fig. \[fig:equilibrium\](a) show that the spin density ${\bf s}_{\rm eq}(x)$ is parallel to ${\bf
M}(x)$, as expected. Similarly, the electron-mediated spin-transfer torque should equal the phenomenological exchange torque density discussed by Brown.[@Brown:1962] This is confirmed by Fig. \[fig:equilibrium\](b), which shows that ${\bf N}_{\rm eq}(x)$ is proportional to ${\bf M}\times {\bf M}''(x)$ for the ${\bf M}(x)$ shown in Fig. \[fig:equilibrium\](a).

![Equilibrium (zero-current) results: (a) Cartesian components of an arbitrarily chosen magnetization ${\bf M}(x)$ (lines); Cartesian components of the calculated spin density ${\bf
  s}_{\rm eq}(x)$ (solid dots); (b) Exchange torque (solid line) and calculated spin-transfer torque (solid dots). []{data-label="fig:equilibrium"}](fig1.eps){width="8cm"}

Semi-classical
--------------

A semi-classical approach to calculating the spin current density is useful for building physical intuition. Accordingly, we write an equation of motion for the spin density of every electron that contributes to the current. This idea has been used in the past, both semi-quantitatively[@Berger:1978] and qualitatively.[@Waintal:2004] Our derivation is based on the behavior of an electron with energy $E$ that moves along the $x$-axis through a uniform magnetic field ${\bf B}_{\rm ex}=B_{\rm ex}\hat{\bf
z}$. The wave function for such 