New identities from quantum-mechanical sum rules of parity-related potentials

TL;DR

This paper explores how quantum sum rules for parity-related potentials reveal new mathematical identities and constraints, especially involving Airy functions and harmonic oscillators, highlighting the impact of potential symmetry on quantum properties.

Contribution

The study introduces novel sum rule constraints for parity-related quantum systems, extending previous work on the quantum bouncer and revealing new relations for the harmonic oscillator.

Findings

Derived new sum rule constraints involving Airy functions.

Discovered novel mathematical relations for parity-restricted harmonic oscillators.

Analyzed the influence of potential symmetry on sum rule convergence.

Abstract

We apply quantum mechanical sum rules to pairs of one-dimensional systems defined by potential energy functions related by parity. Specifically, we consider symmetric potentials, $V(x) = V(-x)$, and their parity-restricted partners, ones with $V(x)$, but defined only on the positive half-line. We extend recent discussions of sum rules for the quantum bouncer by considering the parity-extended version of this problem, defined by the symmetric linear potential, $V(z) = F|z|$ and find new classes of constraints on the zeros of the Airy function, $Ai(z)$, and its derivative $Ai'(z)$. We also consider the parity-restricted version of the harmonic oscillator and find completely new classes of mathematical relations, unrleated to those of the ordinary oscillator problem. These two soluble quantum-mechanical systems defined by power-law potentials provide examples of how the form of the potential (both parity and continuity properties) affects the convergence of quantum-mechanical sum rules. We also discuss semi-classical predictions for expectation values and the Stark effect for these systems.