physics/Optical-transitions.md

@@ -64,7 +64,7 @@ Using the dipole approximation ($`\lambda \gg r`$ or $`\vec{q} \to 0`$) and Eule
 \vec{A}_\text{EM} = \frac{\vec{E}}{2\omega} \left(e^{-i\omega t} + \text{c.c.}\right)
 ```
 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 $`\hbar \omega = \hbar \omega_{fi} = E_f - E_i`$.
 
 ## Matrix elements derived from $`k \cdot p`$ Hamiltonian
 
@@ -101,18 +101,18 @@ We can derive the matrix elements $`v_{fi}`$ of the $`v`$ matrices by using the
 
 ```math
 \begin{split}
-[x,H_0] = \frac{\text{d}H_0}{\text{d}p_x}
+[x,H_0] = i\hbar\frac{\text{d}H_0}{\text{d}p_x}
 \end{split}
 ```
 
-This means that we can get $`v_{x,fi}`$ directly from the matrix reprentation of the $`k \cdot p`$ matrix $`H_{0,fi}`$ by taking the derivation to the respective $`k_x`$ momentum. Note that in the 2D periodic case only $`p_z`$ is an operator (central finite difference) in the Hamiltonian matrix, while $`p_{x,y}`$ just act on the plane wave parts $`e^{i(k_x x + k_y y)}`$ of the wave function and are replaced by $`k_{x,y}`$ momentum values.
+This means that we can get $`v_{x,fi}`$ directly from the matrix reprentation of the $`k \cdot p`$ matrix $`H_{0,fi}`$ by taking the derivation to the respective $`\hbar k_x`$ momentum. Note that in the 2D periodic case only $`p_z`$ is an operator (central finite difference) in the Hamiltonian matrix, while $`p_{x,y}`$ just act on the plane wave parts $`e^{i(k_x x + k_y y)}`$ of the wave function and are replaced by $`\hbar k_{x,y}`$ momentum values.
 
 Therefore, we get
 
 ```math
 \begin{split}
-v_{x,fi} &= \frac{\text{d}H_{0,fi}}{\text{d}k_x}\\
-v_{\pm,fi} &= 2 \frac{\text{d}H_{0,fi}}{\text{d}k_\mp} = \frac{ e E}{\sqrt{2}m\omega_{fi}} \frac{\text{d}H_{0,fi}}{\text{d}k_\mp}
+v_{x,fi} & = \frac{ e E}{2\sqrt{2}m\omega_{fi}} \frac{m}{\hbar}\frac{\text{d}H_{0,fi}}{\text{d}k_x} = \frac{ e E}{2\sqrt{2}\hbar\omega_{fi}} \frac{\text{d}H_{0,fi}}{\text{d}k_x}\\
+v_{\pm,fi} &= \frac{ e E}{2\sqrt{2}m\omega_{fi}} \frac{m}{\hbar} 2 \frac{\text{d}H_{0,fi}}{\text{d}k_\mp} = \frac{ e E}{\sqrt{2}\hbar\omega_{fi}} \frac{\text{d}H_{0,fi}}{\text{d}k_\mp}
 \end{split}
 ```
 where we have used $`\frac{\text{d}}{\text{d}k_\pm} = \frac{\text{d} k_x}{\text{d}k_\pm} \frac{\text{d}}{\text{d}k_x} + \frac{\text{d} k_y}{\text{d}k_\pm} \frac{\text{d}}{\text{d}k_y} = \frac{1}{2}\left(\frac{\text{d}}{\text{d}k_x} \mp i\frac{\text{d}}{\text{d}k_y} \right)`$.