A. Particle in an External Potential: Short Distance Limit
The results obtained in the previous sections for a free particle
are, of course, exact. However,
the same approach can be used to derive the short distance
approximation for the particle's kernel in an arbitrary potential
.
If
is much smaller than the range over which
varies significantly, then we can Taylor expand
in the
neighborhood of
:
(52)
(53)
In this case, the path integral reads
(54)
where
. Integration over the particle trajectory
gives a ``Dirac-delta'' whose argument yields the classical equation of
motion
(55)
which leads, in turn, to the expression for the classical momentum
(56)
Thus, the integration over the momentum trajectory can be defined as
follows
(57)
and the resulting path integral takes the form
(58)
where the normalization constant has been chosen in such a way as to
reproduce equation (36) once the potential is switched off.
.
is much smaller than the range over which
:
![$\displaystyle K( {\vec{\mbox{\boldmath {$x$}}}}- {\vec{\mbox{\boldmath {$x$}}...
...}^{{\vec{\mbox{\boldmath {$x$}}}}}[{\mathcal{D}}y(t)]
[{\mathcal{D}}p(t)]\times$](img129.gif)
![$\displaystyle \quad \times \exp\left\{ {i\over\hbar}\left[
\int_{{\vec{\mbox...
...{$y$}}}}-{\vec{\mbox{\boldmath {$y$}}}}_0) \right) \right]
\right\}
\quad ,$](img130.gif)
. Integration over the particle trajectory
gives a ``Dirac-delta'' whose argument yields the classical equation of
motion
![\begin{displaymath}
\int [{\mathcal{D}}p(t)] \delta\left[ {d{\vec{\mbox{\bold...
...t - {\vec{\mbox{\boldmath {$q$}}}} \right]
\left(\dots\right)
\end{displaymath}](img134.gif){kind=small}
![$\displaystyle \qquad \qquad \times
\int d^3 q \exp\left(-{i T {\vec{\mbox{\...
...x$}}}}_0 +{T^2\over 2\hbar m}
{\vec{\mbox{\boldmath {$F$}}}} \right) \right]$](img136.gif)
![$\displaystyle \qquad \qquad \times
\exp\left[-{i m T\over 2\hbar T}
\left...
...}^2\over 2\hbar m}{\vec{\mbox{\boldmath {$F$}}}} \right)^2 \right]
\quad ,$](img138.gif)
