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

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 $A{i}^{\prime }(z)$. 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.

Published as J. Phys. A: Math. Theor. 43 235202 (22 pp) (2010)

References in corpus (3)

• Measurement of quantum states of neutrons in the Earth's gravitational field
• Quantum mechanical sum rules for two model systems
• The Stark effect in linear potentials

Cited by in corpus (2)

• Exact results for the infinite supersymmetric extensions of the infinite square well
• Less than perfect quantum wavefunctions in momentum-space: How phi(p) senses disturbances in the force