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\section{Quantum Mechanics}

\subsection{Particles and Waves in Classical
Mechanics}

Basic concepts of classical mechanics.

\subsubsection*{Particles}

\begin{definition*}
    Point-particle is an object carrying energy
    $E$ and momentum $p$ in infinitesimally small
    point of space.
\end{definition*}

\noindent
Particle determined by $\bf{x}$ (position) and
$\bf{v} = \dot{\bf{x}} = \dfrac{}{t} \bf{x}$
(velocity). Newton's second law is that
\[ m \ddot{\bf{x}} = \bf{F}(\bf{x}(t),
\dot{\bf{x}}(t)) \]
Solving Newton's second law determines $\bf{x}(t)$
and $\dot{\bf{x}}(t)$ for all $t > t_0$ once
initial conditions known ($\bf{x}(t_0),
\dot{\bf{x}}(t_0)$).

\subsubsection*{Waves}

\begin{definition*}
    Any real or complex-valued function with
    periodicity in time / space.
    \begin{itemize}
        \item Take function of time $t$:
            \[ f(t + T) = f(t) \]
            where $T \neq 0$ is the period.
            \[ \nu = \frac{1}{T} \]
            is the frequency and
            \[ \omega = 2\pi\nu = \frac{2\pi}{T} \]
            is the angular frequency.
        \item Take function of space $x$
            \[ f(x + \lambda) = f(x) \]
            where $\lambda$ is the wavelength.
            \[ K = \frac{2\pi}{\lambda} \]
            is the wave number.
    \end{itemize}
\end{definition*}

\begin{example*}
    In 1 dimension, electromagnetic wave obeys
    equation
    \[ \pfrac[2]{f(x, t)}{t} - c^2 \pfrac[2]{f(x,
    t)}{x} = 0 \tag{1} \]
    with $c \in \RR$. Solutions:
    \[ f_\pm(x, t) = A_\pm \exp(\pm iKx - i\omega t) \]
    with $A_\pm \in \CC$ (amplitude of wave) and
    $\omega = cK$ (dispersion relation), hence
    \[ \lambda = \frac{2\pi c}{\omega} =
    \frac{c}{\nu} \]
\end{example*}

\begin{example*}
    In 3 dimensions, electromagnetic wave obeys
    equation
    \[ \pfrac[2]{f(\bf{x}, t)}{t} - c^2 \nabla^2
    f(\bf{x}, t) = 0 \tag{2} \]
    need $f(x, t_0)$, $\dfrac{f}{t}(x, t_0)$ to
    get unique solution. Solution:
    \[ f(\bf{x}, t) = A\exp(i \bf{K} \cdot \bf{x}
    - i \omega t) \]
    with $\omega = c|\bf{K}|$.
\end{example*}

\begin{note*}
    \begin{itemize}
        \item These kind of waves arise as
            solutions of other governing equations
            provided a different dispersion
            relation.
        \item For all governing equations,
            superposition principle holds if $f_1,
            f_2$ solutions implies $f = f_1 + f_2$
            is a solution.
    \end{itemize}
\end{note*}

\subsection{Particle-like Behaviour of Wave}

\begin{enumerate}
    \item[1.2.I] Black-body Radiation (1900)
    \item[1.2.II] Photo-electric effect (1905)
    \item[1.2.III] Compton scattering (1923)
\end{enumerate}

\subsubsection*{1.2.I Black Body Radiation}

When a body heated at temperature $T$, it radiates
light at different frequencies
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/568d7582456d11ed.png}
\end{center}

Classical prediction:
\[ E = K_B T \]
where $E$ is energy of each wave and $K_B$ is
Boltzmann constant
\[ \implies I(\omega) \propto K_B T
\frac{\omega^2}{\pi^2 c^3} \]
Planck:
\[ I(\omega) \propto \frac{\omega^2}{\pi^2 c^3}
\frac{\hbar \omega}{\exp \left( \frac{\hbar
\omega}{K_B T} \right) - 1} \]
$\hbar$ is reduced Planck constant:
\[ \hbar = \frac{h}{2\pi} \]