[tex] explicit form of \chi_N^\pm; comment on normalization

tex/khodas.tex

@@ -76,19 +76,31 @@ \section{Interface Between Normal and Spin-Orbit Coupling Regions}
 \begin{align}
    \chi_{SO}^{\pm} &= \frac{1}{n_{SO}^{\pm}} 
                       \vect{-p_{x,SO}^{\pm} \pm p_{SO}^\pm}{p_z} \\
-    (n_{SO}^{\pm})^2 &= |-p_{x,SO}^{\pm} \pm p_{SO}^\pm|^2 + p_z^2
+    n_{SO}^{\pm}   &= \sqrt{|-p_{x,SO}^{\pm} \pm p_{SO}^\pm|^2 +
+        p_z^2}
     \label{eq:chi-so-pm}
 \end{align}
 
 where the lower index $x$ means that the value is projected onto the
 $x$ axis. The angle between the $x$ axis and the momentum of the
 incident wave is called $\phi$, so that $p_x = p \cos \phi$.
 
+If one wants to expand $\chi_{SO}^\pm$ in powers of $\ta$, it is
+important to ensure that the normalization condition
+$\chi_{SO}^{\pm\dagger} \cdot \chi_{SO}^\pm$ still holds after the
+expansion. However, the following results have been derived for the
+full (and not expanded) form of $\chi_{SO}^\pm$.
+
 Note that, in the N regime, $H$ is a diagonal matrix, and the direction
 of the eigenvectors can be chosen with some freedom. We pick
 $\chi_N^{\pm} = \lim_{\ta \mapsto 0} \chi_{SO}^{\pm}$ to ensure that
 $<\chi_N^+|\chi_{SO}^+> = 1$ holds true at a vanishing interface.
 
+\begin{align}
+   \chi_N^{\pm} &= \frac{1}{n^{\pm}} 
+                      \vect{-p_x \pm p}{p_z} \\
+    n^{\pm} &= \sqrt{(-p_x \pm p)^2 + p_z^2}
+\end{align}
 
 
 The overall wave function consists of an incident wave, 

tex/summary.tex

@@ -19,14 +19,13 @@ \chapter{Summary and Outlook}
 in $z$-direction.
 
 We also discussed the experimental more accessible setup of two regions with
-different, non-zero strength of spin-orbit interaction, and found that such an
+different, non-zero strengths of spin-orbit interaction, and found that such an
 interface can also be used to achieve some spin polarization, albeit
 of decreasing magnitude when the spin-orbit coupling strengths become similar. 
 
 In both cases a large angle between the incident beam the interface is
 essential for obtaining a decent spin polarization.
 
-
 Future work in this area could involve a four-band model which includes both
 the conductance and valance band for each spin direction, would
 allow more precises modeling of a particular semiconductor, and thus be of

tex/thesis.tex

@@ -183,11 +183,12 @@ \section{Transmission and Green's Functions}
 \end{align}
 
 where $H$ is the Hamiltonian operator. In a one-dimensional wire oriented
-along the $x$ axis, we expect an excitation of the form
+along the $x$ axis with Hamiltonian $H =
+-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$, we expect an excitation of the form
 $\delta = \delta(x - x_0)$ to result in two waves propagating away from $x_0$,
 so our ansatz is
 
-TODO: explicit form of $H$
+% TODO: explicit form of $H$
 
 \begin{align}
     G(x, x_0) = \begin{cases}