physics/Optical-transitions.md

@@ -74,7 +74,7 @@ Using the dipole approximation ($`\lambda \gg r`$ or $`\vec{q} \to 0`$) and Eule
 \end{equation}
 ```
 As in variant 1 we can derive the transition matrix element $`v_{fi}`$.
-The time varying phase factor and its complex conjugate can be identified as the two terms responsible for absorption and stimulated emission (see Fermi's Golden Rule). To fullfill energy conservation, we identify §`\hbar \omega = E_f - E_i`§
+The time varying phase factor and its complex conjugate can be identified as the two terms responsible for absorption and stimulated emission (see Fermi's Golden Rule). To fullfill energy conservation, we identify $`\hbar \omega = E_f - E_i`$.
 
 ## Matrix elements derived from $`k \cdot p`$ Hamiltonian
 
@@ -139,6 +139,37 @@ where we have used $`\frac{\text{d}}{\text{d}k_\pm} = \frac{\text{d} k_x}{\text{
 
 As we already have a symbolic Hamiltonian defintion in `kdotpy`, we can use the method `deriv(to)` to construct suitable symbolic transition matrices which just need to be evaluated for certain Landau Levels (when using `kdotpy-ll`).
 
+With the $`\Gamma`$-point orbital basis 
+
+```math
+\begin{equation}
+\begin{split}
+|1\rangle &= |\Gamma_6, +\tfrac{1}{2}\rangle = |S, \uparrow\rangle\\
+|2\rangle &= |\Gamma_6, -\tfrac{1}{2}\rangle = |S, \downarrow\rangle\\
+|3\rangle &= |\Gamma_8, +\tfrac{3}{2}\rangle =  \tfrac{1}{\sqrt{2}}|X +iY, \uparrow\rangle\\
+|4\rangle &= |\Gamma_8, +\tfrac{1}{2}\rangle =  \tfrac{1}{\sqrt{6}}\left[|X +iY, \downarrow\rangle - 2|Z, \uparrow\rangle\right]\\
+|5\rangle &= |\Gamma_8, -\tfrac{1}{2}\rangle = -\tfrac{1}{\sqrt{6}}\left[|X -iY, \uparrow\rangle + 2|Z, \downarrow\rangle\right]\\
+|6\rangle &= |\Gamma_8, -\tfrac{3}{2}\rangle = -\tfrac{1}{\sqrt{2}}|X -iY, \downarrow\rangle\\
+|7\rangle &= |\Gamma_7, +\tfrac{1}{2}\rangle =  \tfrac{1}{\sqrt{3}}\left[|X +iY, \downarrow\rangle + |Z, \uparrow\rangle\right]\\
+|8\rangle &= |\Gamma_7, -\tfrac{1}{2}\rangle =  \tfrac{1}{\sqrt{3}}\left[|X -iY, \uparrow\rangle - |Z, \downarrow\rangle\right]
+\end{split}
+\end{equation}
+```
+
+the $`8 \times 8`$ Kane Hamiltonian matrix is
+
+```math
+\begin{equation}
+H = 
+\begin{pmatrix}
+\textcolor{blue}{T}  &  0  &  \textcolor{green}{-\tfrac{1}{\sqrt{2}}P k_+}  &    &  \textcolor{green}{\tfrac{1}{\sqrt{6}}P k_-}  &  0  &  -\tfrac{1}{\sqrt{3}}P k_z  &  \textcolor{green}{-\tfrac{1}{\sqrt{3}}P k_-}  \\
+0  &  \textcolor{blue}{T}  &  0  &  \textcolor{green}{-\tfrac{1}{\sqrt{6}}P k_+}  &  \sqrt{\tfrac{2}{3}}P k_z  &  \textcolor{green}{\tfrac{1}{\sqrt{2}}P k_+}  &  \textcolor{green}{-\tfrac{1}{\sqrt{3}}P k_+}  &  \tfrac{1}{\sqrt{3}}P k_z  \\
+\textcolor{green}{-\tfrac{1}{\sqrt{2}}P k_-}  &  0  &  \textcolor{blue}{U+V}  &  \textcolor{green}{-S_-}  &  \textcolor{red}{R}  &  0  &  \textcolor{green}{\tfrac{1}{\sqrt{2}} S_-}  &  \textcolor{red}{-\sqrt{2} R}  \\
+\sqrt{\tfrac{2}{3}}P k_z  &  \textcolor{green}{-\tfrac{1}{\sqrt{6}}P k_-}  &  \textcolor{green}{-S^\dag_-}  &  \textcolor{blue}{U-V}  &  \textcolor{green}{C}  &  \textcolor{red}{R}  &  \textcolor{blue}{\sqrt{2} V}  &  -\sqrt{\tfrac{3}{2}} S_-  
+\end{pmatrix}
+\end{equation}
+```
+ToDo: Unfinished Hamiltonian. For some reason this does not render correctly in some cases.
 
 ## Spectra