Corrected typos

corrected a number of typos that slipped in in the last round of edits

includes/angmomentum-ladder-operators.tex

@@ -25,7 +25,7 @@ \section*{Angular Momentum Raising and Lowering Operators\sectionmark{Exercise:
 
 				\part What are $[\hat M_z,\hat M_+]$ and $[\hat M_z,\hat M_-]$?
 
-					\begin{solution}[2in]
+					\begin{solution}[3in]
 					\end{solution}
 
 			\end{parts}

includes/exchange-symmetry.tex

@@ -12,7 +12,7 @@ \section*{Symmetry with Respect to Exchange\sectionmark{Exercise: Exchange Symme
 
 	\begin{questions}
 
-		\question The permutation operator, $\hat P_{12}$, swaps particles one and 2, e.g. $\hat P_{12} \psi(q_1,q_2) = \psi(q_2,q_1)$.
+		\question The permutation operator, $\hat P_{12}$, swaps particles 1 and 2, e.g. $\hat P_{12} \psi(q_1,q_2) = \psi(q_2,q_1)$.
 
 			\begin{parts}
 				\part If we apply the permutation operator twice, what should we get? In other words, what is $\hat P_{12}^2 \psi(q_1,q_2)$?

includes/matrixmech-eigenstates.tex

@@ -47,7 +47,7 @@ \section*{Eigenvalues, Eigenstates, and Eigenvectors\sectionmark{Exercise: Eigen
 					\begin{solution}[1.25in]
 					\end{solution}	
 
-				\part Now, assume we only know the matrix elements of $\hat H$ in some other basis, $\{\ket i\}$.  What are the matrix elements in this basis (i.e. what is $H_{ij}$?
+				\part Now, assume we only know the matrix elements of $\hat H$ in some other basis, $\{\ket i\}$.  What are the matrix elements in this basis (i.e. what is $H_{ij}$)?
 
 					\begin{solution}[1.25in]
 					\end{solution}	

includes/matrixmech-intro.tex

@@ -41,7 +41,7 @@ \section*{Describing States and Operators with Matrices\sectionmark{Exercise: In
 
 		\question 
 		\begin{parts}
-			\part If we calculate matrix elements of $\hat A$ using one basis set (e.g. the particle-in-a-box eigenstates), will it be the same as if we use a different basis set (e.g. the harmonic oscillator eigenstates)?  Why or why not?
+			\part If we calculate matrix elements of $\hat A$ using one basis set (e.g. the particle-in-a-box eigenstates), will they be the same as if we use a different basis set (e.g. the harmonic oscillator eigenstates)?  Why or why not?
 
 				\begin{solution}[2.5in]
 				\end{solution}

includes/perturbation-first-order.tex

@@ -33,7 +33,7 @@ \section*{First-Order Perturbation Theory\sectionmark{Exercise: First-Order Pert
 				\begin{solution}[1.25in]
 				\end{solution}
 
-			\part Because the equation from part (a) must hold for \emph{any} value of $\lambda$, the terms that scale as $\lambda$ on the  left side must be equal to the ones that scale as $\lambda$ on the right side $\lambda \neq 0$ (the same would be true for the terms that scale as $\lambda^2$, but we haven't included all of those, so we'll ignore them for now).
+			\part Because the equation from part (a) must hold for \emph{any} value of $\lambda$, the terms that scale as $\lambda$ on the  left side must be equal to the ones that scale as $\lambda$ on the right side (the same would be true for the terms that scale as $\lambda^2$, but we haven't included all of those, so we'll ignore them for now).
 
 			If you include only the terms that have exactly one factor of $\lambda$, what equation results?
 

includes/the-H-atom.tex

@@ -16,7 +16,7 @@ \section*{The Schr\"odinger Equation for the Hydrogen Atom\sectionmark{Exercise:
 
 		\question A ``hydrogen-like'' atom consists of a nucleus of charge $+Ze$ and mass $m_n$ and an electron of charge $-e$ and mass $m_e$.
 			\begin{parts}
-				\part If the nucleus is at coordinates $\vec{R_n} = (x_n,y_n,z_n)$ and the electron is at $\vec{R_e} = (x_e,y_e,z_e)$, what is the total kinetic energy for this system?
+				\part If the nucleus is at coordinates $\vec{R_n} = (x_n,y_n,z_n)$ and the electron is at $\vec{R_e} = (x_e,y_e,z_e)$, what is the total kinetic energy operator for this system?
 
 				\emph{Hint: you can simplify the notation by remembering that $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$.  You can give $\nabla^2$ a subscript to indicate which set of coordinates (nuclear or electronic) it is operating on.}
 

includes/the-He-atom.tex

@@ -8,7 +8,7 @@
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\section*{Motivation: The Helium Atom\sectionmark{Exercise: The Helium Atom}}
+\section*{The Helium Atom\sectionmark{Exercise: The Helium Atom}}
 
 	\begin{questions}
 

includes/variational-principle.tex

@@ -10,10 +10,10 @@
 
 \section*{The Variational Principle\sectionmark{Exercise: The Variational Principle}}
 
-	In this set of exercises, we'll consider the value of the expectation value of the energy for an arbitrary state.  For these exercises, assume that the $\{\ket{n}\}$ are energy eigenstates, with $\hat H \ket{n} = E_n \ket{n}$.
+	In this set of exercises, we'll consider the expectation value of the energy for an arbitrary state, $\ket{\Psi_{trial}}$.  For these exercises, assume that the $\{\ket{n}\}$ are energy eigenstates, with $\hat H \ket{n} = E_n \ket{n}$.
 
 \begin{questions}
-	\question Suppose we pick a trial state $\ket{\Psi_{trial}} = \sum_n c_n \ket{n}$.
+	\question Suppose we pick $\ket{\Psi_{trial}}$ such that $\ket{\Psi_{trial}} = \sum_n c_n \ket{n}$.
 
 		\begin{parts}
 			\part What is the expectation value of the energy, $E_{trial}$, for this state?