Eqs. (27)–(28) — Gaussian moments are finite, Lorentzian moments diverge

Let $\psi(x)=\sqrt{e^{-x^2}/\sqrt\pi}$ and $\phi(y)=1/\sqrt{\pi(y^2+1)}$. Then

1. $\displaystyle\int_{\mathbb R}\psi(x)^2\,dx=1$ and $\displaystyle\int_{\mathbb R}\phi(y)^2\,dy=1$;
2. for every $n\in\mathbb N$, $\displaystyle\int_{\mathbb R}|x|^n\,\psi(x)^2\,dx<\infty$;
3. for every $n\ge2$, $\displaystyle\int_{\mathbb R}|y|^n\,\phi(y)^2\,dy=\infty$.

The Gaussian state has finite position moments of all orders, while the Lorentzian state has infinite position moments from the second on (in particular infinite $\langle X^2\rangle$).

Formalization Note Finiteness of $\int g$ for a nonnegative $g$ is expressed as Lebesgue integrability of $g$, and infiniteness as its failure.