ass2: Finish 5, add TODOs

superconductivity_assignment2_kvkempen.tex

@@ -29,6 +29,8 @@
 \usepackage{float}
 \usepackage{mathtools}
 \usepackage{amsmath}
+\usepackage{todonotes}
+\setuptodonotes{inline}
 
 \newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
 
@@ -68,6 +70,7 @@ giving free energy
 	\mathcal{F}_0(T \leq T_c) = \frac{-a^2}{\beta}(T-T_c)^2 + \frac{a^2}{2\beta}(T-T_c)^2 = \frac{-a^2}{2\beta}(T-T_c)^2 \leq \mathcal{F}_0(T \geq T_c)
 \]
 where we chose the positive of the $\pm$ as the order parameter is understood to increase from finite at the phase transition.
+\todo{Is this a reasonable statement? It actually does not really matter that much as mostly $\psi^2$ is used, but the physical meaning is totally different. It implies some kind of symmetry, too. It seems that also \cite{abrikosov} mentions this.}
 	
 For the specific heat, we find
 \[
@@ -163,7 +166,6 @@ Next, we can equate the previously found supercurrent for our foil to the Ginzbu
 as $A_y \perp \hat{y}$, giving zero partial derivative.
 
 In our case, indeed the rigid gauge choice gives the criterium for the London gauge ($\nabla \cdot \vec{A} = 0$).
-\end{enumerate}
 
 In the rigid gauge, the order parameter $\psi$ is constant in space and time.
 To then also have that $\nabla \cdot \vec{A} = 0$, follows from the expression for the supercurrent as we saw earlier.
@@ -185,8 +187,20 @@ This is the case for $\nabla \cdot \vec{J_s}$, or, in words, when there is no co
 If this is not the case (if the divergence is non-zero), there is conversion between normal current and supercurrent.
 This result seems to Waldram's conclusion in \cite[p. 24--26]{waldram}.
 
+\item
+We apply a gauge transformation as follows.
+\begin{align}
+	\chi(\vec{r}, t) &= \frac{-\hbar}{2e}(\omega t - \vec{k} \cdot \vec{r}) \\
+	\vec{A} &\to \vec{A} + \nabla\chi = \vec{A} + \frac{\hbar}{2e} \vec{k} \\
+	\phi &\to \phi - \pfrac{\chi}{t} = \phi + \frac{\hbar}{2e} \omega
+\end{align}
+
+\todo{Do I really need to put in the previously found $\vec{A}$?}
+\end{enumerate}
+
 \section{Type II superconductors and the vortex lattice}
 
+
 \section{Currents inside type-II superconducting cylinder}
 
 \bibliographystyle{vancouver}