Normal and anti-normal ordered expressions for annihilation and creation operators

TL;DR

This paper derives normal and anti-normal ordered forms of the number operator raised to power k and provides general formulas for functions of the number operator in these orders, enhancing operator manipulation techniques.

Contribution

It introduces explicit normal and anti-normal order expressions for powers of the number operator and general functions, based on commutation relations, advancing operator ordering methods.

Findings

Explicit formulas for normal and anti-normal ordered powers of the number operator.

General expressions for functions of the number operator in normal and anti-normal order.

Enhanced techniques for manipulating creation and annihilation operator functions.

Abstract

We give the normal and anti-normal order expressions of the number operator to the power $k$ by using the commutation relation between the annihilation and creation operators. We use those expressions to give general formulae for functions of the number operator in normal and anti-normal order.