Divergence of $\langle p^6\rangle$ in discontinuous potential wells

TL;DR

This paper proves that the expectation value of the sixth power of momentum diverges in potential wells with finite jump discontinuities, extending known results from square wells to a broader class of half-potential wells.

Contribution

It demonstrates the divergence of <p^6> in potential wells with finite jump discontinuities, generalizing previous findings beyond square wells.

Findings

<p^6> diverges in square and half-potential wells with discontinuities

The divergence occurs for potentials expressed as V(x)=-U(x)Θ(x) with specific properties

The results apply to a broad class of discontinuous potential wells.

Abstract

The surprising divergence of the expectation value $<\!p^6\!>$ for the square well potential is known. Here, we prove and demonstrate the divergence of $<\!p^6\!>$ in potential wells which have a finite jump discontinuity; apart from the square-well two-piece half-potentials wells are examples. These half-potential wells can be expressed as $V(x)=-U(x) \Theta(x)$, where $\Theta(x)$ is the Heaviside step function. $U(x)$ are continuous and differentiable functions with minimum at $x=0$ and which may or not vanish as $x\sim \infty$.