# New approach for deriving operator identities by alternately using normally, antinormally, and Weyl ordered integration

## TL;DR

This paper introduces a novel method using alternating operator orderings—normal, antinormal, and Weyl—to derive new operator identities and integration formulas related to special functions, bridging quantum operator techniques with mathematical analysis.

## Contribution

It presents a new approach for deriving operator identities and integration formulas by alternately applying different quantum operator orderings, expanding the tools available in quantum mechanics and mathematical physics.

## Key findings

- Derived new operator ordering identities.
- Established integration formulas for Laguerre and Hermite polynomials.
- Proposed a novel method linking quantum operator orderings with mathematical integration.

## Abstract

Dirac's ket-bra formalism is the "language" of quantum mechanics and quantum field theory. In Refs.(Fan et al, Ann. Phys. 321 (2006) 480; 323 (2008) 500) we have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors. In this work by alternately using the technique of integration within normal, antinormal, and Weyl ordering of operators we not only derive some new operator ordering identities, but also deduce some useful integration formulas regarding to Laguerre and Hermite polynomials. This opens a new route of deriving mathematical integration formulas by virtue of the quantum mechanical operator ordering technique.