This may be a possible errata but Sakurai (pp 126 in the 2nd Edition) states that starting with
$$S = \int dt \,\,\scr{L_{\mathrm{classical}}}$$
Looking at a finite-time-interval of the action:

\begin{equation}
S(n,n-1) = \int\limits_{t_{n-1}}^{t_n} dt \left( \frac{m \dot{x}^2}{2} - V(x)\right)\,\,\,\,\,\,(1)
\end{equation}

Now using a straight-line approximation for points $(x_{n-1},t_{n-1})$ and $(x_n,t_n)$, which implies

$$\dot{x} = \frac{x_n-x_{n-1}}{t_n-t_{n-1}} = \frac{x_n-x_{n-1}}{\Delta t}$$
Sakurai then has Eq (1) become:
$$S(n,n-1) = \Delta t\left( \left(\frac{m}{2}\right) \left(\frac{x_n-x_{n-1}}{\Delta t}\right)^2 - V\left(\frac{x_n+x_{n-1}}{2}\right)\right)$$

My question is why is the potential now dependent on $ V\left(\frac{x_n + x_{n-1}}{2}\right) \mathrm{\,\,instead \,\,\,of\,\,\,} V\left(x_n-x_{n-1}\right)$?

I checked the most recent errata (4/5/13) posted by J. Napolitano ( http://homepages.rpi.edu/~napolj/ErrataMQM.pdf ) and there is none for this page. Could anyone clarify this step for me? Thanks.

This post imported from StackExchange Physics at 2014-04-15 16:40 (UCT), posted by SE-user John M