For one dimensional conductors, for example, the scattering state arising from a unitary flux of particles at energy $E$ originating in the $i$-th reservoir reads $$\varphi_i(x)=\frac{e^{ik_i(E)x}+r_i(E) e^{-ik_i(E)x}}{\sqrt{hv_i(E)}} ~,$$ for the $i$-th lead, and $$\varphi_j(x)=\frac{t_{ji}(E) e^{-ik_j(E)x}}{\sqrt{hv_j(E)}} ~,$$ for the $j$-th lead, with $j\ne i$. Here $r_i(E)$ is the reflection amplitude for particles at energy $E$, wave vector $k_i(E)$ and group velocity $v_i(E)$ and $t_{ji}(E)$ is the transmission amplitude from lead $i$ to lead $j$. Note that $|r_i|^2$ is the probability for a particle to reflect back into the $i$-th lead and $|t_{ji}|^2$ is the probability for the transmission of a particle from lead $i$ to lead $j$. In the second quantization formalism, the field operator $\hat{\psi}_{j\sigma}(x,t)$ for spin $\sigma$ particles in lead $j$ is built from scattering states and it is defined as [@lesovik] $$\hat{\psi}_{j\sigma}(x,t)= \int dE ~ \frac{e^{-\frac{iEt}{\hbar}}}
{\sqrt{hv_j(E)}}
\left[ \hat{a}_{j\sigma}(E)e^{ik_j x}+\hat{\phi}_{j\sigma}(E)e^{-ik_j x}
\right] ~,$$ where $\hat{a}_{j\sigma}(E)$ ($\hat{\phi}_{j\sigma}(E)$) is the destruction operator for incoming (outgoing) particles at energy $E$ with spin $\sigma$ in lead $j$. These operators are linked by the equation $$\left( \begin{array}{c} \hat{\phi}_{1\uparrow}\\ \hat{\phi}_{1\downarrow} \\
\hat{\phi}_{2\uparrow} \\ \vdots
\end{array} \right)=~S~
\left( \begin{array}{c} \hat{a}_{1\uparrow}\\ \hat{a}_{1\downarrow} \\
\hat{a}_{2\uparrow} \\ \vdots
\end{array} \right)
\label{Smat}$$ and obey anti-commutation relations $$\left\{ \hat{a}^{\dagger}_{i\sigma}(E),~\hat{a}_{j\sigma'}(E')\right\}
=\delta_{i,j} 