Expectation values of $p^2$ and $p^4$ in the square well potential

TL;DR

This paper derives simple analytic expressions for the expectation values of $p^2$ and $p^4$ in a finite square well potential, revealing the finite moments only for these specific powers due to the momentum distribution's fall-off.

Contribution

It provides a novel analytic approach to compute $<p^2>$ and $<p^4>$ in the finite square well, overcoming derivative discontinuity issues and analyzing the momentum distribution's decay.

Findings

Analytic expressions for $<p^2>$ and $<p^4>$ obtained

Momentum distribution $(p)$ falls off as $p^{-6}$

Expectation values are finite only for $s=2,4$

Abstract

Position and momentum representations of a wavefunction $\psi(x)$ and $\phi(p)$, respectively are physically equivalent yet mathematically in a given case one may be easier or more transparent than the other. This disparity may be so much so that one has to device a special strategy to get the quantity of interest in one of them. We revisit finite square well (FSW) in this regard. Circumventing the the problems of discontinuity of second and higher derivatives of $\psi(x)$ we obtain simple analytic expressions of $<\!p^2\!>$ and $<\!p^4\!>$. But it is the surprising fall-off of $\phi(p)$ as $p^{-6}$ that reveals and restricts $<\!p^s\!>$ to be finite and non-zero only for $s=2,4$. In finding $<\!p^s\!>(s=2,4)$ from $\phi(p)$, $p$-integrals are improper which for time-being, have been evaluated numerically to show the agreement between two representations.