Top-hat wavefunction ($\psi (x)=L^{-1/2}{1}_{|x|<L/2}$)

A top-hat wavefunction is constant on an interval and zero elsewhere. It has a uniform position distribution and a sinc function momentum amplitude. Its discontinuous edges place it outside the finite-kinetic-energy domain of the free Hamiltonian. Smooth-edged localized packets can approximate its position profile while having different momentum tails.

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.