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\title{Quantum Mechanics (very messy notes)}
\author{A. Student}
\date{}

\begin{document}
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\maketitle

\section{Schr\"odinger Equation}
The time-dependent Schr\"odinger equation\cacographynote*{maybe add time-indep. case here?} is
\begin{equation}
  i\hbar \frac{\partial}{\partial t}\psi(\mathbf{r},t)
  = -\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf{r},t) + V(\mathbf{r},t)\psi(\mathbf{r},t)
\end{equation}
For a stationary state we can write $\psi(\mathbf{r},t) = \phi(\mathbf{r})e^{-iEt/\hbar}$,
which gives
\begin{equation}
  -\frac{\hbar^2}{2m}\nabla^2\phi(\mathbf{r}) + V(\mathbf{r})\phi(\mathbf{r}) = E\phi(\mathbf{r}).
\end{equation}

\section{Some remarks}
Note:
\begin{itemize}
  \item $V(\mathbf{r})$ can be complicated.
  \item Boundary conditions matter!
  \item Check normalization: $\int |\psi|^2 \, d^3r = 1$.
  \item Stationary $\Rightarrow$ $|\psi|^2$ does not change with time.
\end{itemize}

\end{document}