deletions | additions       diff --git a/photoemission2.tex b/photoemission2.tex  index 9486ef3..7fb4b0b 100644  --- a/photoemission2.tex  +++ b/photoemission2.tex 
...
\begin{array}{l}  \varphi^A_i(\vec{r},t+\Delta t) = M \hat{U}(\Delta t) \phi^A_i(\vec{r},t)\ ,\\  \varphi^B_i(\vec{r},t+\Delta t) =  \frac{M}{(2\pi)^{\frac{3}{2}}}\int \frac{M}{(2\pi)^{3/2}}\int  {\rm d}\vec{k} e^{\mathrm{i}\vec{k}\cdot\vec{r}} \hat{U}_{\rm v}(\Delta t) \phi^B_i(\vec{k},t) \ ,\\  \vartheta^A_i(\vec{k},t+\Delta t) =  \frac1{(2\pi)^{\frac{3}{2}}} \frac1{(2\pi)^{3/2}}  \int {\rm d}\vec{r} e^{-\mathrm{i}\vec{k}\cdot\vec{r}} (1-M) \hat{U}(\Delta t) \phi^A_i(\vec{r},t) \ ,\\  \vartheta^B_i(\vec{k},t+\Delta t) =  \hat{U}_{\rm v}(\Delta t) \phi^B_i(\vec{k},t) -   \frac1{(2\pi)^{\frac{3}{2}}} \frac1{(2\pi)^{3/2}}  \int {\rm d}\vec{r} e^{-\mathrm{i}\vec{k}\cdot\vec{r}} \varphi^B_i(\vec{r},t+\Delta t)\ .  \end{array}  \end{align} 
...
It is the time-evolution operator associated with the Hamiltonian $\hat{H}_{\rm v}$ describing free electrons in   an oscillating field.   Given a time dependent vector field $\vec{{A}}(t)$, the Hamiltonian   $\hat{H}_{\rm v}=\frac{1}{2}(-i\vec{\nabla}-\frac{\vec{{A}}(t)}{c})^2$ v}=\frac{1}{2}\left(-i\vec{\nabla}-\frac{\vec{{A}}(t)}{c}\right)^2$  expressed in the velocity gauge is diagonal in momentum and can be naturally applied to $\phi^B_i(\vec{k},t)$.  For all systems that can be described by a Hamiltonian such that $\hat{H}(\vec{r},t)=\hat{H}_{\rm v}(\vec{r},t)$