is to study the extent to which the spin-transfer torque associated with real domain wall configurations satisfies Eq. (\[eqn:adia\]).

Quantum {#sec:quantum}
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In light of Eq. (\[eqn:magnetization\]), the exchange magnetic field that the enters the Hamiltonian in Eq. (\[eqn:Hamiltonian\]) is $$\label{eqn:exchangeB}
{\bf B}_{\rm  ex}(x) =B_{\rm ex}{\left( \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta \right)}.$$ For a given energy, the eigenfunctions for a conduction electron with wave-vector ${\bf k}$ take the form $\Psi_\pm({\bf r},{\bf
k})=\psi_\pm(x, k_x)e^{ik_y y+ ik_z z}$ where the spinor $\psi_\pm(x,k_x)$ satisfies $${\left[ -{d^2\over dx^2} - k_B^2{\left( 
    \begin{array}[c]{cc}
    \cos\theta & e^{-i\phi}\sin\theta \\
    e^{i\phi}\sin\theta & -\cos\theta 
    \end{array}  \right)}  \right]}\psi_\pm
    = \kappa_\pm^2\psi_\pm.
    \label{eqn:Schrodinger}$$ In this expression, $$\label{eqn:kdef}
\kappa_\pm^2=k_x^2\mp k_B^2,$$ and $\pm$ refers to majority/minority band electrons. The single-electron spin density and spin current density are[@Stiles:2002] $$\label{eqn:sesd}
{\bf s}_\pm(x,k_x)={\hbar \over 2}\sum_{\alpha,\beta}
\psi^\ast_{\pm,\alpha}(x,k_x)\bm{\sigma}_{\alpha,\beta}\psi_{\pm,\beta}(x,k_x)$$ and $$\label{eqn:sescd}
{\bf Q}_\pm(x,k_x)=-{\hbar^2 \over 2m}\sum_{\alpha,\beta} {\rm Im} [
  \psi^\ast_{\pm,\alpha}(x,k_x)\bm{\sigma}_{\alpha,\beta}{d \over dx}
  \psi_{\pm,\beta}(x,k_x)].$$

As a check, we used this formalism to calculate the equilibrium (zero applied current) spin density ${\bf s}_{\rm eq}(x)$ and equilibrium spin current density ${\bf Q}_{\rm eq}(x)$ for a magnetization distribution chosen arbitrarily except for the constraint that $|{\bf
M}(x)|$ be uniform. The densities ${\bf s}_{\rm eq}(x)$ and ${\bf
Q}_{\rm eq}(x)$ are obtained by retaining only the second term in square 