= Solution
Split the <time-evolution operator> into $N$ factors with $\epsilon=T/N$, and insert $N-1$ <position> <completeness relations> between them. Each factor is evaluated by inserting <momentum> states normalized by $\langle x|p\rangle=(2\pi\hbar)^{-1/2}e^{ipx/\hbar}$:
$$
\begin{aligned}
\langle x_{j+1}|e^{-i\epsilon p^2/(2m\hbar)}|x_j\rangle
&=\int_{\mathbb R}\frac{dp}{2\pi\hbar}
\exp\left[\frac{ip(x_{j+1}-x_j)}{\hbar}
-\frac{i\epsilon p^2}{2m\hbar}\right]\\
&=\sqrt{\frac{m}{2\pi i\hbar\epsilon}}\,
\exp\left[\frac{im(x_{j+1}-x_j)^2}{2\hbar\epsilon}\right].
\end{aligned}
$$
Multiplying the $N$ kernels and integrating the intermediate <positions> produces the stated time-sliced <path integral>, including its $N/2$ normalization power. For a <free particle> this <Gaussian> short-time kernel is exact, so the convolution identity already holds at finite $N$. Oscillatory <integrals> are defined with a vanishing convergence regulator, consistently fixing the square-root phase.
Back to article page
