Momentum variance of a discontinuous top hat ($|\tilde{\psi }(p){|}^2=\frac{L}{2\pi \hslash }{\mathrm{sinc}}^2(pL/(2\hslash ))$)

The momentum density of a top-hat wavefunction decays as an oscillatory $1/p^2$ envelope. Its momentum second moment therefore diverges, although the probability density itself is normalizable. The central diffraction-lobe width is not the standard deviation. Even the first moment requires care: the symmetric principal value is zero, but the absolute first moment diverges logarithmically.