Quantum mechanical sum rules for two model systems

TL;DR

This paper explores quantum mechanical sum rules through examples involving the infinite well and delta-function potential, demonstrating their applications and confirming their mathematical consistency, while also calculating Stark effect energy shifts.

Contribution

It provides explicit examples of quantum sum rules in simple systems and shows how to verify them using different mathematical techniques, including energy shift calculations.

Findings

Sum rules are applicable to the infinite well and delta-function potential.

Different mathematical methods can confirm sum rules.

Second-order Stark effect energy shifts are evaluated.

Abstract

Sum rules have played an important role in the development of many branches of physics since the earliest days of quantum mechanics. We present examples of one-dimensional quantum mechanical sum rules and apply them in two familiar systems, the infinite well and the single delta-function potential. These cases illustrate the different ways in which such sum rules can be realized, and the varying mathematical techniques by which they can be confirmed. Using the same methods, we also evaluate the second-order energy shifts arising from the introduction of a constant external field, namely the Stark effect.