Commutation relations for functions of canonical conjugate operators

TL;DR

This paper derives a general formula for the commutator of functions of canonical conjugate operators, extending applicability to functions like the Coulomb potential that are not expressible as Taylor series.

Contribution

It introduces a novel formalism for computing commutators of operator functions, valid for infinite series of positive and negative powers, including non-analytic functions.

Findings

Derived explicit formulas for operator function commutators

Extended applicability to non-analytic functions like Coulomb potential

Formalism reduces to classical Poisson brackets in certain limits

Abstract

In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum mechanics, first term in the sum is formally equivalent to Poisson bracket in classical mechanics, which is a well-known result. The novelty respect other papers is the type of functions of operators considered: equations mentioned are proved valid when each function is an infinite series of positive and negative powers of these operators, as long as every series converges. Therefore, this formalism can be applied to functions such as the Coulomb potential, where the inverse of radial distance cannot be expressed as a Taylor series.