Divergence of $⟨p^6⟩$ in 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 $V(x)=-U(x)\hat{I}˜(x)$, where $\hat{I}˜(x)$ is the Heaviside step function. $U(x)$ are continuous and differentiable functions with minimum at $x=0$ and which may or not vanish as $x\sim \infty$.

Five figures and 5 pages