= Momentum variance of a discontinuous top hat
{title2=$|\widetilde\psi(p)|^2=\frac{L}{2\pi\hbar}\operatorname{sinc}^2(pL/(2\hbar))$}
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.
Back to article page
