 Solution

Use the unitary position-to-momentum Fourier transform $\varphi (p)=(2\pi \hslash {)}^{-1/2}\int \psi (x)e^{-ipx/\hslash }dx$. Integration giveswhere $\mathrm{sinc}u=\sin u/u$ with its continuous value one at zero. The amplitude is real and even, with central maximum $\sqrt{a/(2\pi \hslash )}$, zeros at $p=2n\pi \hslash /a$ for nonzero integer $n$, and alternating-sign sidelobes decaying as $1/|p|$.

The probability density is $|\varphi {|}^2$, not the signed Sinc function amplitude. Its integral is one by the unitarity of the Fourier transform.