Less than perfect quantum wavefunctions in momentum-space: How phi(p) senses disturbances in the force

TL;DR

This paper presents a systematic method to analyze the asymptotic behavior of the momentum-space wavefunction in one-dimensional quantum systems, linking potential discontinuities to power-law tails in phi(p).

Contribution

It introduces a general approach connecting potential discontinuities to the large |p| behavior of phi(p), extending understanding of wavefunction tails in quantum mechanics.

Findings

Discontinuities in derivatives of the potential induce specific power-law tails in phi(p).

The order of the potential's derivative discontinuity determines the tail's decay rate.

The approach is validated through familiar quantum examples.

Abstract

We develop a systematic approach to determine the large |p| behavior of the momentum-space wavefunction, phi(p), of a one-dimensional quantum system for wich the position-space wavefunction, psi(x), has a discontinuous derivative at any order. We find that if the k-th derivative of the potential energy function has a discontinuity, there is a corresponding discontinuity in psi^{(k+2)}(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wavefunction proportional to 1/p^{k+3}. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.