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\newpage
\section{Foundation of Quantum Mechanics}

\begin{center}
    \begin{tabular}{c|c}
        Linear Algebra & Quantum Mechanics \\
        vector ($n$-dimensional complex value) &
        state \\
        $\bf{v}$, $\{e_i\}$, $\bf{v} \to (v_1,
        \dots, v_n)$ & $\psi$, basis $\bf{x} \to
        \psi(\bf{x}, t)$ \\
        vector space $\CC^n$ & $L^2(\RR^3)$
        complex-valued square integrable functions
        \\
        inner product $\langle \bf{v}, \bf{w}
        \rangle = v_1^* w_1 + \cdots + v_n^* w_n$ &
        $\langle \psi, \phi \rangle \int_{\RR^3}
        \psi^* (\bf{x}, t) \phi(\bf{x}, t) \dd^3
        x$ \\
        linear map $\CC^n \to \CC^n$, use matrix &
        $L^2(\RR^3) \to l^2(\RR^3)$ operators
        $\hat{O}$, $\phi = \hat{O} \psi$
    \end{tabular}
\end{center}

\subsection{Wave Function and Probabilistic
Interpretation}

Classical mechanics: $\bf{x}, \dot{\bf{x}}$ (or
equivalently $\bf{p} = m \dot{\bf{x}}$) determine
dynamics of the particle. \myskip
Quantum mechanics: $\psi$ described by
$\psi(\bf{x}, t)$ determine dynamics of the
particle (in a probabilistic way)

\begin{flashcard}[state-of-particle]
\begin{definition*}
    $\psi$ is the \cloze{\emph{state} of the
    particle.}
\end{definition*}
\end{flashcard}

\begin{definition*}
    $\psi(\bf{x}, t)$ complex coefficient of
    $\psi$ in the continuous basis of $\bf{x}$,
    i.e. $\psi(\bf{x}, t)$ is $\psi$ in $\bf{x}$
    representation and is called
    \emph{wavefunction}. $\psi(\bf{x}, t) : \RR^3
    \to \CC$ that satisfies mathematical
    properties dictated by its physics
    interpretation.
\end{definition*}

\subsubsection*{Interpretations}

Born's rule / probabilistic interpretation.
\myskip
The probability density for particle to sits at
$\bf{x}$ at given time $t$
\[ \rho(\bf{x}, t) \propto |\psi(\bf{x}, t)|^2 \]
$\rho(\bf{x}, t) \dd V$ is the probability that
the particle sits in some small volume $V$ centred
at $\bf{x}$ is proportional to square modulus of
$\psi(\bf{x}, t)$.

\subsubsection*{Mathematical Properties}

\begin{enumerate}[(i)]
    \item Because the particle has to be somewhere
        implies that wavefunction has to be
        normalisable (or square0integrable) in
        $\RR^3$:
        \[ \int_{\RR^3} \psi^*(\bf{x}, t)
        \psi(\bf{x}, t) \dd^3 x = \int_{\RR^3}
        |\psi(\bf{x}, t)|^2 \dd^3 x = \mathcal{N}
        < \infty \]
        with $\mathcal{N} \in \RR$ and
        $\mathcal{N} \neq 0$.
    \item Because total probability has to be 1,
        \[ \ol{\psi}(\bf{x}, t) =
        \frac{1}{\sqrt{\mathcal{N}}} \psi(\bf{x},
        t) \]
        \[ \implies \int_{\RR^3}
        |\ol{\psi}(\bf{x}, t)|^2 \dd^3 x = 1 \]
        \[ \implies \rho(\bf{x}, t) =
        |\ol{\psi}(\bf{x}, t)|^2 \]
\end{enumerate}

\begin{note*}
    Often drop $\ol{\psi}$ and write wavefunctions
    as $\psi$, then normalise at the end.
\end{note*}

\begin{note*}
    If $\tilde{\psi}(\bf{x}, t) = e^{i\alpha}
    \psi(\bf{x}, t)$ with $\alpha \in \RR$ then
    $|\tilde{\psi}(\bf{x}, t)|^2 = |\psi(\bf{x},
    t)|^2$ so $\psi$ and $\tilde{\psi}$ are
    equivalent state.
\end{note*}

\noindent
Non-examinable aside: \\
State corresponds to rays in vector space of wave
functions $[\psi]$ is the equivalence class of
vectors under equivalence relation $\psi_1 \sim
\psi_2 \iff \psi_1 = e^{i\alpha} \psi_2$.

\subsubsection*{Hilbert Space}

\begin{flashcard}[square-integrable-hilbert-space]
\begin{definition*}
    The set of all \cloze{square-integrable
    functions} in
    $\RR^3$ is called \cloze{Hilbert space
    $\mathcal{H}$}
    or \cloze{$L^2(\RR^3)$.}
\end{definition*}
\end{flashcard}

\begin{proposition*}
    If $\psi_1, \psi_2 \in \mathcal{H}$ then $\psi
    = a_1 \psi_1 + a_2 \psi_2 \neq 0 \in
    \mathcal{H}$ ($a_1, a_2 \in \CC$).
\end{proposition*}

% theorem 2.1
\begin{theorem}
    If $\psi_1(\bf{x}, t)$ and $\psi_2(\bf{x}, t)$
    are square-integrable then also $\psi(\bf{x},
    t) = a_1 \psi_1(\bf{x}, t) + a_2
    \psi_2(\bf{x}, t)$ is square-integrable.
\end{theorem}

\begin{proof}
    \[ \int_{\RR^3} |\psi_1(\bf{x}, t)|^2 \dd^3 x =
    \mathcal{N}_1 < \infty \]
    \[ \int_{\RR^3} |\psi_2(\bf{x}, t)|^2 \dd^3 x =
    \mathcal{N}_2 < \infty \]
    by triangle identities for complex numbers,
    \begin{align*}
        \int_{\RR^3} |\psi(\bf{x}, t)|^2 \dd^3 x
        &= \int_{\RR^3} |a_1 \psi_1(\bf{x}, t) +
        a_2 \psi_2(\bf{x}, t)|^2 \dd^3 x \\
        &\le \int_{\RR^3} (|a_1 \psi_1(\bf{x}, t)| +
        |a_2 \psi_2 (\bf{x}, t)|)^2 \dd^3 x \\
        &= \int_{\RR^3} (|a_1 \psi_1(\bf{x}, t)|^2
        + |a_2\psi_2(\bf{x}, t)|^2 +
        2|a_1\psi_1||a_2\psi_2|) \dd^3 x \\
        &\le \int_{\RR^3} 2|a_1\psi_1(\bf{x},
        t)|^2 + 2|a_2 \psi_2(\bf{x}, t)|^2 \dd^3 x
        \\
        &= 2|a_1|^2 \mathcal{N}_1 + 2|a_2|^2
        \mathcal{N}_2 \\
        &< \infty
    \end{align*}
\end{proof}

\subsection{Inner Product}

\begin{flashcard}[hilbert-inner-product]
\begin{definition*}
    Inner product in $\mathcal{H}$ is defined as
    \[ \langle \psi, \phi \rangle = \cloze{\int_{\RR^3}
    \psi^*(\bf{x}, t)
    \phi(\bf{x}, t) \dd^3 x} \]
\end{definition*}
\end{flashcard}

\begin{theorem}
    If $\psi, \phi \in \mathcal{H}$ then their
    inner product is guaranteed to exist.
\end{theorem}

\begin{proof}
    \[ \int_{\RR^3} |\psi(\bf{x}, t)|^2 \dd^3 x =
    \mathcal{N}_1 < \infty \]
    \[ \int_{\RR^3} |\phi(\bf{x}, t)|^2 \dd^3 x =
    \mathcal{N}_2 < \infty \]
    \begin{align*}
        |\langle \psi, \phi \rangle |
        &= \left| \int_{\RR^3} \psi^*(\bf{x}, t)
        \phi(\bf{x}, t) \dd^3 x \right| \\
        &\le \sqrt{\int_{\RR^3} |\psi(\bf{x},
        t)|^2 \dd^3 x \int_{\RR^3} |\phi(\bf{x},
        t)|^2 \dd^3 x}
        &&\text{(Cauchy Schwarz)} \\
        &= \sqrt{\mathcal{N}_1 \mathcal{N}_2} \\
        &< \infty \qedhere
    \end{align*}
\end{proof}
