Clean up projective measurements section

src/thesis/chapters/2_fundamentals.tex

@@ -635,20 +635,6 @@ We will represent the state of a particle with wave function
 $\psi(x,t)$ using the vector $\ket{\psi}$
 \cite[Sec.~3.3]{griffiths_introduction_1995}.
 
-% We can model a wave function $\psi(x,t)$ as a linear combination of different
-% \emph{basis functions} $e_n(x,t),~n\in \mathbb{N}$ as%
-% \begin{align*}
-%     \psi(x,t) = \sum_{n=1}^{\infty} c_n \cdot e_n(x,t)
-%     .%
-% \end{align*}
-% To express this relation using linear algebra, we represent
-% $\psi(x,t)$ and $e_n(x,t)$ as vectors $\ket{\psi}$ and $\ket{e_n}$.
-% We write%
-% \begin{align*}
-%     \ket{\psi} = \sum_{n=1}^{\infty}  c_n \ket{e_n}
-%     .%
-% \end{align*}
-
 % Operators
 
 Another important notion is that of an \emph{operator}, a transformation
@@ -785,7 +771,7 @@ We can decompose an arbitrary state as $\ket{\psi} = \sum_{n=1}^{\infty} c_n
 \ket{e_n}$, where $\lvert c_n \rvert ^2$ represents the probability
 of obtaining a certain measurement value.
 Note that when we speak of an \emph{observable}, we are usually
-refering to the corresponding operator $\hat{Q}$.
+refering to the operator $\hat{Q}$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Projective Measurements}
@@ -793,24 +779,23 @@ refering to the corresponding operator $\hat{Q}$.
 
 % Projective measurements
 
-% TODO: Introduce concept of collapsing the wave function onto a basis state
 The measurements we considered in the previous section, for which
 \autoref{eq:gen_expr_Q_exp_lin} holds, belong to the category of
 \emph{projective measurements}.
-For these, certain restrictions such as repeatability apply: after
-measuring a quantum state and thus collapsing it onto one of the
-determinate states, futher measurements should yield the same value.
+For these, certain restrictions such as repeatability apply: the act
+of measuring a quantum state should \emph{collapse} it onto one of
+the determinate states.
+Further measurements should then yield the same value.
 More general methods of modelling measurements exist, e.g., describing
-destructive measurements, but they are not relevant to us here
-\cite[Box~2.5]{nielsen_quantum_2010}.
+destructive measurements \cite[Box~2.5]{nielsen_quantum_2010}, but
+they are not relevant to us here.
 
 % Projection operators
 
-% TODO: Fix notational issues related to e_n
 We can model the collapse of the original state onto one of the
 superimposed basis states as a \emph{projection}.
-To see this, we insert \autoref{eq:determinate_basis} into
-\autoref{eq:observable_eigenrelation}, obtaining%
+To see this, we use Equations \ref{eq:determinate_basis} and
+\ref{eq:observable_eigenrelation} to compute
 \begin{align*}
     \hat{Q}\ket{\psi} = \sum_{n=1}^{\infty} c_n \hat{Q} \ket{e_n}
     = \sum_{n=1}^{\infty} \lambda_n c_n \ket{e_n}
@@ -823,7 +808,7 @@ the separate components as
 \begin{align*}
     \hat{Q} = \sum_{n=1}^{\infty} \lambda_n \hat{P}_n
 \end{align*}
-using \emph{projection operators}
+using \emph{projection operators} \cite[Eq.~3.160]{griffiths_introduction_1995}
 \begin{align*}
     \hat{P}_n := \ket{e_n}\bra{e_n}, \hspace{3mm} n\in \mathbb{N}
     .
@@ -839,10 +824,12 @@ A particularly interesting property of projection operators is that
     = \hat{P}_n \ket{\psi},
 \end{align*}%
 and the only way this can hold for any $\ket{\psi}$ is if $\hat{P}_n$
-only has the eigenvalues $0$ or $1$.
-As explained in the previous section, the eigenvalues are the results
-of performing a measurement.
-We can thus use the projection operator as an observable and treat
+only has the eigenvalues $0$ or $1$
+% tex-fmt: off
+\cite[Prob.~3.57a)]{griffiths_introduction_1995}.
+% tex-fmt: on
+The eigenvalues can again be interpreted as possible measurement results.
+We can thus use the $\hat{P}$ as an observable and treat
 the eigenvalue as an indicator of the state having a component along
 the related basis vector.