diff --git a/.gitignore b/.gitignore
index 5fd6a19..7b5dc27 100644
--- a/.gitignore
+++ b/.gitignore
@@ -4,3 +4,10 @@ PS2/doc.fls
 PS2/doc.log
 PS2/doc.synctex.gz
 PS2/doc.synctex.gz
+PS3/doc.aux
+PS3/doc.fdb_latexmk
+PS2/doc.synctex.gz
+PS3/doc.fls
+PS3/doc.log
+PS3/doc.synctex.gz
+PS2/doc.synctex.gz
diff --git a/PS2/doc.synctex.gz b/PS2/doc.synctex.gz
index a2bb6d0..e6dd941 100644
Binary files a/PS2/doc.synctex.gz and b/PS2/doc.synctex.gz differ
diff --git a/PS3/doc.pdf b/PS3/doc.pdf
index fc911b2..fb1e818 100644
Binary files a/PS3/doc.pdf and b/PS3/doc.pdf differ
diff --git a/PS3/doc.tex b/PS3/doc.tex
index 6fc32e5..2ebef0a 100644
--- a/PS3/doc.tex
+++ b/PS3/doc.tex
@@ -180,7 +180,7 @@ S_u = [1, s; s, 1];
             \end{enumerate}
         \end{enumerate}
     \newpage
-    
+
     \section*{Problem 2}
         \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] 
             \item %2a
@@ -194,7 +194,7 @@ S_u = [1, s; s, 1];
                         \includegraphics[width=0.5\textwidth]{q2ai.png}
                     }
                     % TODO still don't like this caption
-                    \captionof{figure}{Energy vs. wave vector relationship (note: not discretized to account for N)}
+                    \captionof{figure}{Energy vs. wave vector relationship.}
                 \end{minipage}
             
                 \item %2aii
@@ -208,6 +208,14 @@ S_u = [1, s; s, 1];
                 \begin{equation*}
                     \{ \phi \} = C e ^{ i k \cdot n a }
                 \end{equation*}
+                \begin{align*}
+                    \left\{\phi\right\} = \begin{bmatrix}
+                                            Ce^{ika} \\
+                                            Ce^{ik2a} \\
+                                            \vdots \\
+                                            C_Ae^{ikNa}
+                                            \end{bmatrix} 
+                \end{align*}
 
                 The corresponding wave function is:
 
@@ -230,7 +238,7 @@ S_u = [1, s; s, 1];
                                             C_Be^{ik2a} \\
                                             \vdots \\
                                             C_Ae^{ikNa} \\
-                                            C_Ae^{ikNa}
+                                            C_Be^{ikNa}
                                             \end{bmatrix} 
                 \end{align*}
                 \item %2bii