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\def\Re{\mathop{\rm Re}\nolimits}

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\noindent PHY--396~L.\quad
Problem set \#13.\quad
Due January 27, 2005.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
First, a reading assignment:
{\it Quantum Mechanics and Path Integrals} by Feynman \& Hibbs.
Read all you can about care and use of Path Integrals.

\medskip
\point
Second, a generalized reading assignment:
Refresh your memories of Statistical Mechanics,
especially partition functions and related quantities.
Use any graduate-level SM book you like.

\bigskip
\point
Finally, an actual exercise.
Consider a particle on a 1D circle of radius $R$, or equivalently
a 1D particle in a box of length $L=2\pi R$ with periodic boundary
conditions where moving past the $X=L$ points brings you back to $x=0$.
In other words, the particle's position $x(t)$ is defined modulo $L$.

\spointbegin
Consider all possible particle's paths from a fixed point $x_0$ (modulo~$L$)
at time $t=0$ to a fixed point $x'$ (modulo~$L$) at time $t=T$.
Show that the space of such paths is isomorphic to the space of free
particle's paths from a fixed $x_0$ at $t=0$ to any of the points $x'+nL$
at $t=T$, for all integer $n=0,\pm1,\pm2,\ldots$.
Then use path integral formalism to show that
$$
U_{\rm box}(x';x_0)\
=\,\sum_{n=-\infty}^{+\infty} U_{\rm free}(x'+nL;x_0)
\eqn\BoxFree
$$
where $U_{\rm box}$ and $U_{\rm free}$ are the evolution kernels (between times
$t=0$ and $t=T$) for the particle in a box and for the free particle.

\pointcon
According to Poisson re-summation formula, if a function $F(n)$ of integer $n$
can be continued to a function $F(\nu)$ of arbitrary real $\nu$, then
$$
\eqalign{
\sum_{n=-\infty}^{+\infty} F(n)\ &
=\int\!\!d\nu\,F(\nu)\times\sum_{n=-\infty}^{+\infty}\delta(\nu-n)\cr
&=\,\sum_{\ell=-\infty}^{+\infty}
 \int\!\!d\nu\,F(\nu)\times e^{2\pi i\ell\nu}.\cr
}\eqn\Poisson
$$

\spoint
The free particle (living on an infinite 1D line) has evolution kernel
$$
U_{\rm free}(x';x_0)\ =\ \sqrt{M\over 2\pi i\hbar T}\times
    \exp\left(+{iM(x'-x_0)^2\over 2\hbar T}\right).
\eqn\Free
$$
Plug this free kernel into eq.~\BoxFree\ and use Poisson formula to sum
over~$n$.

\spoint
Verify that the resulting evolution kernel for the particle in a box
agrees with the usual QM formula
$$
U_{\rm box}(x';x_0)\
=\,\sum_p L^{-1/2}e^{ipx'/\hbar}\times e^{-iT(p^2/2M)/\hbar}
	\times L^{-1/2}e^{-ipx_0/\hbar}
\eqno\eq
$$
where  $p$ takes box-quantized values
$$
p\ =\ {2\pi\hbar\over L}\times\rm integer.
\eqno\eq
$$

\bye



