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\begin{document}

\title{How To Be A Grumpy Fuzzball in 3 Days}
\author{Jessica Not-A-Physics Major\\
\small MIT Department of Physics\\%[-0.25in] 
\small Cambridge, MA 02142\\[-0.25in]}
\date{} 

\maketitle

\thispagestyle{empty}

\begin{abstract}
\noindent We present the path integral formulation of quantum
mechanics and demonstrate its equivalence to the Schr\"odinger
picture. We apply the method to the free particle and quantum harmonic
oscillator, investigate the Euclidean path integral, and discuss other
applications.
\end{abstract}

\section{Introduction}

Before perusing the really hairy examples below, look at some examples of some basic math typsetting to get us oriented:\\

% fractions
This is an inline fraction: $\frac{1}{2}$ = $\frac{x}{4}$\\

% subscripts, superscripts
These are also inline: H$_{2}$O$_{2}$ has subscripts and cm$^{-1}$ has a superscript\\

How about a square root? $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$\\

Using \textit{equation} numbers your equation and displayes it centered on a line by itself. Take a
look at the difference when you do and don't escape text in math mode:

\begin{equation}
f(x)= \psi\times x for all integers
\end{equation}

\begin{equation}
f(x)= \psi\times x \mbox{ for all integers} 
\label{eq2}
\end{equation}

The label above lets you reference the equation elsewhere, saving you copy-and-paste time. If you don't care for a numbered equation, you can use brackets:\\

\[ f(x)= \psi\times x \mbox{ for all integers} \]

\section{Introduction}

A fundamental question in quantum mechanics is \textit{how does the state
of a particle evolve with time}? That is, the determination the
time-evolution $\ket{\psi(t)}$ of some initial state
$\ket{\psi(t_0)}$. Quantum mechanics is fully
predictive in the sense that initial conditions and
knowledge of the potential occupied by the particle is enough to fully
specify the state of the particle for all future times.\footnote{In
the analysis below, we consider only the position of a particle, and
not any other quantum property such as spin.}

\[ f(x)=\frac{1}{2} - x \]

\begin{equation}
\ket{\psi(t)} = \sum_{n=0}^{n=\infty}\exp\left[-iE_nt/\hbar\right]\braket{n}{\psi(t_0)}\ket{n}
\label{ufrome}
\end{equation}

In the early twentieth century, Erwin Schr\"odinger derived an
equation specifies how the instantaneous change in the wavefunction
$\frac{d}{dt}\ket{\psi(t)}$ depends on the system inhabited by the
state in the form of the Hamiltonian. In this formulation, the
eigenstates of the Hamiltonian play an important role, since their
time-evolution is easy to calculate (i.e. they are stationary). A
well-established method of solution, after the entire eigenspectrum of
$\hat{H}$ is known, is to decompose the initial state into this
eigenbasis, apply time evolution to each and then reassemble the
eigenstates. That is,

\begin{equation}
\ket{\psi(x,t')} =
\int_{-\infty}^{\infty}\braket{\psi(x',t')}{\psi(x_0,t_0)}dx'\ket{\psi(x',t')}
\label{ufromprop}
\end{equation}

Far from the classical trajectory, the rapidly oscillating terms
in~(\ref{ufromprop}) can cause convergence issues and are generally
unpleasant to deal with. Observe that in Minkowski spacetime with one
physical coordinate, the proper distance $\sigma^2$ goes as the
negative square of the time $-t^2$. However, in Euclidean spacetime,
the sign of the $t^2$ term is positive; they differ by a phase factor
$i$. Consider what would happen if we introduced a factor of $i$ into
the exponentials - the oscillating terms would turn into decaying
exponentials, which have an entirely different physical meaning.

\section{Path Integral Method}

Define the {\em propagator} of a quantum system between two spacetime
points $(x',t')$ and $(x_0,t_0)$ to be the probability transition
amplitude between the wavefunction evaluated at those points. 

\begin{equation}
U(x',t';x_0,t_0) = \braket{\psi(x',t')}{\psi(x_0,t_0)}
\label{firstu}
\end{equation}
If the Hamiltonian carries no explicit time-dependence, we can relabel
the first time-value $t_0 = 0$ and work only with elapsed time $t = t'
- t_0$. We will often write~(\ref{firstu}) as $U(x',t;x_0)$ to
illustrate this. The propagator above, along with an initial state
ket, fully describes the evolution of a system over time. It is also
customary, as is done in Sakurai~\cite{sakurai}, to use here the
symbol $K$ instead of $U$ and refer to~\label{prop} as the ``kernel''
or ``Feynman kernel''. The path integral method, as we are about to
see, is an explicit way to construct this propagator.\footnote{These are excerpts from an 8.09 paper I stole from dvp. Merci Dennis!}

\begin{equation}
U(x',t;x_0) = A(t)\sum_{\mbox{all
trajectories}}\exp\left[\frac{i}{\hbar}S[x(t)]\right]\label{ufrompath}
\end{equation}

\newpage

\subsection*{Acknowledgments}

\begin{thebibliography}{9}

\bibitem{hibbs} R.P. Feynman and A.R. Hibbs, {\sl Quantum Mechanics
and Path Integrals} (McGraw-Hill, New York, 1965)

\bibitem{sakurai} J. J. Sakurai, {\sl Modern Quantum Mechanics}
(Addison-Wesley, Reading, MA, 1994)

\bibitem{shankar} R. Shankar, {\sl Principles of Quantum Mechanics,
2nd Ed.} (Plenum Press, New York, NY, 1994)

\bibitem{feynman} R. Feynman, {\sl Space-Time Approach to
Non-Relativistic Quantum Mechanics} (1948) Rev. Modern Physics. 20

\bibitem{mackenzie} R. MacKenzie, {\sl Path Integral Methods and
Applications} (2000) {\tt arXiv:quant-ph/0004090v1}

\bibitem{grosche} C. Grosche, {\sl An Introduction Into the Feynman
Path Integral} (1993) {\tt arXiv:hep-th/9302097v1}

\end{thebibliography}

\end{document}