form $$|P_k\rangle =|k,maj\rangle + B|-k,maj\rangle
\label{p}$$ in the polariser and $$|L_k\rangle =F|k,\uparrow\rangle
\label{p2}$$ in the lead. A majority spin in either ferromagnet enters or leaves the spacer without scattering, since in our simple model there is no potential step. Also the minority spin wave-function entering a ferromagnet is zero. The spacer wave-function may therefore be written in two ways: $$|S_k\rangle =F|k,\uparrow\rangle+E \left({\rm e}^{-{\rm
i}k(N+1)a}|k,\downarrow\rangle- {\rm e}^{{\rm
i}k(N+1)a}|-k,\downarrow\rangle \right) 
\label{pp}$$ or $$\begin{aligned}
\label{ppp}
|S_k\rangle &=& |k,maj\rangle+B|-k,maj\rangle+D\left(|k,min\rangle-
|-k,min\rangle\right)\nonumber\\
&=& \cos\left(\psi/2\right)|k,\uparrow\rangle+\sin\left(\psi/2\right)
|k,\downarrow\rangle
+B\left[\cos\left(\psi/2\right)|-k,\uparrow\rangle+\sin\left(\psi/2\right)|-k,\downarrow\rangle\right]\\
&&+D\left[-\sin\left(\psi/2\right)|k,\uparrow\rangle+\cos\left(\psi/2\right)
|k,\downarrow\rangle+\sin\left(\psi/2\right)
|-k,\uparrow\rangle-\cos\left(\psi/2\right)|-k,\downarrow\rangle\right].\nonumber\end{aligned}$$ On equating coefficients of $|k,\uparrow\rangle$, $|k,\downarrow\rangle$, $|-k,\uparrow\rangle$, $|-k,\downarrow\rangle$ in expressions (\[pp\]) and (\[ppp\]) we have four equations which may be solved for $B$, $D$, $E$, $F$. In particular the transmission coefficient $T$ is given by $$\label{T}
T=\left|F\right|^2=\frac{4\cos^2(\psi/2)\sin^2
  k(N+1)a}{\sin^4(\psi/2)+
4\cos^2(\psi/2)\sin^2k(N+1)a}.$$ Similarly an electron incident from the right with wave-function $|-k\uparrow\rangle$ in the lead is partially reflected and finally emerges as a partially transmitted wave $F^{\prime}|-k,maj\rangle$ in the polarising magnet. It is found that $F^{\prime}=F$ so that the transmission coefficient is the same for particles from left or right.

The spin angular momentum current in a particular layer, which we shall denote by S although it need not be the spacer layer, is the sum of currents carried by 