physics/Optical-transitions.md

 # Optical transitions from $`k \cdot p`$ matrix elements
 
+This article derives the calculatoin of optical transitions and spectral plots generated from those transitions. We will start out from very basic assumptions.
+
+[[_TOC_]]
+
 ## Ansatz: Perturbation Hamiltonian for EM wave
 
+Starting out from the $`k \cdot p`$ Hamiltonian describing out sample (in static electric and magnetic fields), we introduce the interaction with electro-magnetic (EM) waves by another perturbative ansatz.
+
+There are two variants for this ansatz common in literature.
+
+### Variant 1: Atomistic approach
+
+We derive the perturbation Hamiltonian by considering the EM wave as an additional potential for the atom core and the electron, such that $`H = H_0 +H_1`$ with
+
+```math
+\begin{equation}
+\begin{split}
+H_1 &= - e \phi(\vec{r}_e) + q \phi(\vec{r}_c) \\
+    &= e\left[\phi(\vec{r}_c) - \phi(\vec{r}_e)\right] + (q-e)\phi(\vec{r}_c)
+\end{split}
+\end{equation}
+```
+where $`\phi`$ is the electric potential of the EM wave.
+
+Since the last term only describes the atomic core, which is not of interesst in our case, we can neglect it.\
+For the first term we use a Taylor series up to first order in the electron-core distance $`\vec{r}`$ for the electric potential. As the EM wave potential varies on length scales much larger then the electron-core distance, we can neglect higher orders. This is known as the dipol approximation. Thus we can write
+
+```math
+\begin{equation}
+\begin{split}
+H_1 &= - e \vec{r} \cdot \vec{\nabla}\phi(\vec{r}_c) \\
+    &= - \vec{\mu}_e \cdot \vec{E}(\vec{r}_c)
+\end{split}
+\end{equation}
+```
+where $`\vec{\mu}_e`$ is an effective electric dipole moment.
+
+Since the commutator $`[H_0,H_1]`$ does not vanish, both Hamiltonians do not have the same eigenstates and $`H_1`$ will introduce transitions between our known eigenstates of $`H_0`$. The strength of such a transition is given by the transition matrix element
+
+```math
+\begin{equation}
+\begin{split}
+v_{fi} &= \langle \psi_f | H_1 | \psi_i \rangle \\
+       &= e \vec{E} \langle \psi_f | \vec{r} | \psi_i \rangle \\
+       &= \frac{-i \hbar e \vec{E}}{(E_f - E_i)m} \langle \psi_f | \vec{p} | \psi_i \rangle
+\end{split}
+\end{equation}
+```
+
+We can use the above commutator relation to [show](https://en.wikipedia.org/wiki/Transition_dipole_moment#In_terms_of_momentum) the last identity. Note that this identity is only true, as long as there is only the $`p^2/2m`$ term in the Hamiltonian. Upon inclusion of a vector potential, there will also be a term linear in $`\vec{A} \cdot \vec{p}`$. In our case this is negelectable, as this will in the end not contribute to transitions but just act as homogenoues energy offset for all states.
+
+### Variant 2: Vector potential
+
+Let's write our Hamiltonian as 
+
+```math
+\begin{equation}
+\begin{split}
+H &= \frac{(\vec{p} +e \vec{A})^2}{2m_0} +V_c\\
+  &= \frac{(\vec{p} +e \vec{A}_0 +e \vec{A}_\text{EM})^2}{2m_0} +V_c\\
+=H_0 + H_\text{EM} &= \left[\frac{(\vec{p} +e \vec{A}_0)^2}{2m_0} +V_c\right] + \left[\frac{e \vec{A}_\text{EM} \cdot\vec{p}}{m_0}\right]
+\end{split}
+\end{equation}
+```
+where we have split the vector potential $`A`$ into the static part $`A_0`$ of the electric and magnetic fields applied to the `kdotpy` state calculation and $`\vec{A}_\text{EM} = \frac{\vec{E}}{\omega} \cos(\omega t - \vec{q}\cdot\vec{r})`$ for the EM wave. We used the identity $`\vec{p} \cdot \vec{A} = \vec{A} \cdot \vec{p}`$ valid for divergence free fields of EM waves and negelected the energy offset terms $`\vec{A}^2_\text{EM} + 2 \vec{A}_\text{0}\cdot\vec{A}_\text{EM}`$. The latter term is by the way the same term missing in the $`\vec{r} \leftrightarrow\vec{p}`$ relation above in [Variant 1](#variant-1-atomistic-approach).
+
+Using the dipole approximation ($`\lambda \gg r`$ or $`\vec{q} \to 0`$) and Euler's formula we can rewrite 
+
+```math
+\begin{equation}
+\vec{A}_\text{EM} = \frac{\vec{E}}{2\omega} \left(e^{-i\omega t} + \text{c.c.}\right)
+\end{equation}
+```
+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).
+
 ## Matrix elements derived from $`k \cdot p`$ Hamiltonian
 
+TBF: Use p := [r,H] := dH/dk
+
 ## Spectra
 
 In order to compare $`k \cdot p`$ results with experimental data, we calculate energy spectra for measureable quantities (absorption, rotation, ellipticity).