= Solution
Immediately after a uniformly illuminated hard slit, the normalized transmitted state is $\psi(x)=w^{-1/2}$ for $|x|<w/2$, zero elsewhere. With Fourier convention $\widetilde\psi(p)=(2\pi\hbar)^{-1/2}\int e^{-ipx/\hbar}\psi(x)dx$,
$$
\boxed{\widetilde\psi(p_x)=\sqrt{\frac{w}{2\pi\hbar}}\operatorname{sinc}\frac{p_xw}{2\hbar},\quad \widetilde p(p_x)=\frac1{\delta p_x}\operatorname{sinc}^2\frac{\pi p_x}{\delta p_x},\quad\delta p_x=\frac hw.}
$$
The supplied sinc <integral> verifies unit <normalization>. The first zeros occur at $\pm\delta p_x$, but this finite central-lobe width is not the rms uncertainty: $p_x^2\widetilde p$ fails to have a convergent <integral>.
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